what is the experimental probability of rolling an even number

Experimental Probability

The chance or occurrence of a particular event is termed its probability. The value of a probability lies between 0 and 1 which means if it is an impossible event, the probability is 0 and if it is a certain event, the probability is 1. The probability that is determined on the basis of the results of an experiment is known as experimental probability. This is also known as empirical probability.

What is Experimental Probability?

Experimental probability is a probability that is determined on the basis of a series of experiments. A random experiment is done and is repeated many times to determine their likelihood and each repetition is known as a trial. The experiment is conducted to find the chance of an event to occur or not to occur. It can be tossing a coin, rolling a die, or rotating a spinner. In mathematical terms, the probability of an event is equal to the number of times an event occurred ÷ the total number of trials. For instance, you flip a coin 30 times and record whether you get a head or a tail. The experimental probability of obtaining a head is calculated as a fraction of the number of recorded heads and the total number of tosses. P(head) = Number of heads recorded ÷ 30 tosses.

Experimental Probability Formula

The experimental probability of an event is based on the number of times the event has occurred during the experiment and the total number of times the experiment was conducted. Each possible outcome is uncertain and the set of all the possible outcomes is called the sample space. The formula to calculate the experimental probability is: P(E) = Number of times an event occurs/Total number of times the experiment is conducted

Consider an experiment of rotating a spinner 50 times. The table given below shows the results of the experiment conducted. Let us find the experimental probability of spinning the color - blue.

The experimental probability of spinning the color blue = 10/50 = 1/5 = 0.2 = 20%

Experimental Probability vs Theoretical Probability

Experimental results are unpredictable and may not necessarily match the theoretical results. The results of experimental probability are close to theoretical only if the number of trials is more in number. Let us see the difference between experimental probability and theoretical probability.

Experimental Probability Examples

Here are a few examples from real-life scenarios.

a) The number of cookies made by Patrick per day in this week is given as 4, 7, 6, 9, 5, 9, 5.

Based on this data, what is the reasonable estimate of the probability that Patrick makes less than 6 cookies the next day?

P(< 6 cookies) = 3/7 = 0.428 = 42%

b) Find the reasonable estimate of the probability that while ordering a pizza, the next order will not be of a pepperoni topping.

Based on this data , the reasonable estimate of the probability that the next type of toppings that would get ordered is not a pepperoni will be 15/20 = 3/4 = 75%

Related Sections

• Card Probability
• Conditional Probability Calculator
• Binomial Probability Calculator
• Probability Rules
• Probability and Statistics

Important Notes

• The sum of the experimental probabilities of all the outcomes is 1.
• The probability of an event lies between 0 and 1, where 0 is an impossible event and 1 denotes a certain event.
• Probability can also be expressed in percentage.

Examples on Experimental Probability

Example 1: The following table shows the recording of the outcomes on throwing a 6-sided die 100 times.

Find the experimental probability of: a) Rolling a four; b) Rolling a number less than four; c) Rolling a 2 or 5

Experimental probability is calculated by the formula: Number of times an event occurs/Total number of trials

a) Rolling a 4: 17/100 = 0.17

b) Rolling a number less than 4: 56/100 = 0.56

c) Rolling a 2 or 5: 31/100 = 0.31

Example 2: The following set of data shows the number of messages that Mike received recently from 6 of his friends. 4, 3, 2, 1, 6, 8. Based on this, find the probability that Mike will receive less than 2 messages next time.

Mike has received less than 2 messages from 2 of his friends out of 6.

Therefore, P(<2) = 2/6 = 1/3

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Practice Questions on Experimental Probability

Frequently asked questions (faqs), how do you find the experimental probability.

The experimental probability of an event is based on actual experiments and the recordings of the events. It is equal to the number of times an event occurred divided by the total number of trials.

What is the Experimental Probability of rolling a 6?

The experimental probability of rolling a 6 is 1/6. A die has 6 faces numbered from 1 to 6. Rolling the die to get any number from 1 to 6 is the same and the probability (of getting a 6) = Number of favorable outcomes/ total possible outcomes = 1/6.

What is the Difference Between Theoretical and Experimental Probability?

Theoretical probability is what is expected to happen and experimental probability is what has actually happened in the experiment.

Do You Simplify Experimental Probability?

Yes, after finding the ratio of the number of times the event occurred to the total number of trials conducted, the fraction which is obtained is simplified.

Which Probability is More Accurate, Theoretical Probability or Experimental Probability?

Theoretical probability is more accurate than experimental probability. The results of experimental probability are close to theoretical only if the number of trials are more in number.

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Experimental Probability

Here we will learn about experimental probability, including using the relative frequency and finding the probability distribution.

There are also probability distribution worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is experimental probability?

Experimental probability i s the probability of an event happening based on an experiment or observation.

To calculate the experimental probability of an event, we calculate the relative frequency of the event.

We can also express this as R=\frac{f}{n} where R is the relative frequency, f is the frequency of the event occurring, and n is the number of trials of the experiment.

If we find the relative frequency for all possible events from the experiment we can write the probability distribution for that experiment.

The relative frequency, experimental probability and empirical probability are the same thing and are calculated using the data from random experiments. They also have a key use in real-life problem solving.

For example, Jo made a four-sided spinner out of cardboard and a pencil.

She spun the spinner 50 times. The table shows the number of times the spinner landed on each of the numbers 1 to 4. The final column shows the relative frequency.

The relative frequencies of all possible events will add up to 1.

This is because the events are mutually exclusive.

Step-by-step guide: Mutually exclusive events

Experimental probability vs theoretical probability

You can see that the relative frequencies are not equal to the theoretical probabilities we would expect if the spinner was fair.

If the spinner is fair, the more times an experiment is done the closer the relative frequencies should be to the theoretical probabilities.

In this case the theoretical probability of each section of the spinner would be 0.25, or \frac{1}{4}.

Step-by-step guide: Theoretical probability

How to find an experimental probability distribution

In order to calculate an experimental probability distribution:

Draw a table showing the frequency of each outcome in the experiment.

Determine the total number of trials.

Write the experimental probability (relative frequency) of the required outcome(s).

Explain how to find an experimental probability distribution

Experimental probability worksheet

Get your free experimental probability worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on   probability distribution

Experimental probability  is part of our series of lessons to support revision on  probability distribution . You may find it helpful to start with the main probability distribution lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Probability distribution
• Relative frequency
• Expected frequency

Experimental probability examples

Example 1: finding an experimental probability distribution.

A 3 sided spinner numbered 1,2, and 3 is spun and the results recorded.

Find the probability distribution for the 3 sided spinner from these experimental results.

A table of results has already been provided. We can add an extra column for the relative frequencies.

2 Determine the total number of trials

3 Write the experimental probability (relative frequency) of the required outcome(s).

Divide each frequency by 110 to find the relative frequencies.

Example 2: finding an experimental probability distribution

A normal 6 sided die is rolled 50 times. A tally chart was used to record the results.

Determine the probability distribution for the 6 sided die. Give your answers as decimals.

Use the tally chart to find the frequencies and add a row for the relative frequencies.

The question stated that the experiment had 50 trials. We can also check that the frequencies add to 50.

Divide each frequency by 50 to find the relative frequencies.

Example 3: using an experimental probability distribution

A student made a biased die and wanted to find its probability distribution for use in a game. They rolled the die 100 times and recorded the results.

By calculating the probability distribution for the die, determine the probability of the die landing on a 3 or a 4.

The die was rolled 100 times.

We can find the probability of rolling a 3 or a 4 by adding the relative frequencies for those numbers.

P(3 or 4) = 0.22 + 0.25 = 0.47

Example 4: calculating the relative frequency without a known frequency of outcomes

A research study asked 1200 people how they commute to work. 640 travelled by car, 174 used the bus, and the rest walked. Determine the relative frequency of someone not commuting to work by car.

Writing the known information into a table, we have

We currently do not know the frequency of people who walked to work. We can calculate this as we know the total frequency.

The number of people who walked to work is equal to

1200-(640+174)=386.

We now have the full table,

The total frequency is 1200.

Divide each frequency by the total number of people (1200), we have

The relative frequency of someone walking to work is 0.321\dot{6} .

How to find a frequency using an experimental probability

In order to calculate a frequency using an experimental probability:

Multiply the total frequency by the experimental probability.

Explain how to find a frequency using an experimental probability

Example 5: calculating a frequency

A dice was rolled 300 times. The experimental probability of rolling an even number is \frac{27}{50}. How many times was an even number rolled?

An even number was rolled 162 times.

Example 6: calculating a frequency

A bag contains different coloured counters. A counter is selected at random and replaced back into the bag 240 times. The probability distribution of the experiment is given below.

Determine the number of times a blue counter was selected.

As the events are mutually exclusive, the sum of the probabilities must be equal to 1. This means that we can determine the value of x.

1-(0.4+0.25+0.15)=0.2

The experimental probability (relative frequency) of a blue counter is 0.2.

Multiplying the total frequency by 0.1, we have

240 \times 0.2=48.

A blue counter was selected 48 times.

Common misconceptions

• Forgetting the differences between theoretical and experimental probability

It is common to forget to use the relative frequencies from experiments for probability questions and use the theoretical probabilities instead. For example, they may be asked to find the probability of a die landing on an even number based on an experiment and the student will incorrectly answer it as 0.5.

• The relative frequency is not an integer

The relative frequency is the same as the experimental probability. This value is written as a fraction, decimal or percentage, not an integer.

Practice experimental probability questions

1. A coin is flipped 80 times and the results recorded.

Determine the probability distribution of the coin.

As the number of tosses is 80, dividing the frequencies for the number of heads and the number of tails by 80, we have

2. A 6 sided die is rolled 160 times and the results recorded.

Determine the probability distribution of the die. Write your answers as fractions in their simplest form.

Dividing the frequencies of each number by 160, we get

3. A 3 -sided spinner is spun and the results recorded.

Find the probability distribution of the spinner, giving you answers as decimals to 2 decimal places.

Dividing the frequencies of each colour by 128 and simplifying, we have

4. A 3 -sided spinner is spun and the results recorded.

Find the probability of the spinner not landing on red. Give your answer as a fraction.

Add the frequencies of blue and green and divide by 128.

5. A card is picked at random from a deck and then replaced. This was repeated 4000 times. The probability distribution of the experiment is given below.

How many times was a club picked?

6. Find the missing frequency from the probability distribution.

The total frequency is calculated by dividing the frequency by the relative frequency.

Experimental probability GCSE questions

1. A 4 sided spinner was spun in an experiment and the results recorded.

(a) Complete the relative frequency column. Give your answers as decimals.

(b) Find the probability of the spinner landing on a square number.

Total frequency of 80.

2 relative frequencies correct.

All 4 relative frequencies correct 0.225, \ 0.2, \ 0.3375, \ 0.2375.

Relative frequencies of 1 and 4 used.

0.4625 or equivalent

2. A 3 sided spinner was spun and the results recorded.

Complete the table.

Process to find total frequency or use of ratio with 36 and 0.3.

3. Ben flipped a coin 20 times and recorded the results.

(a) Ben says, “the coin must be biased because I got a lot more heads than tails”.

Comment on Ben’s statement.

(b) Fred takes the same coin and flips it another 80 times and records the results.

Use the information to find a probability distribution for the coin.

Stating that Ben’s statement may be false.

Mentioning that 20 times is not enough trials.

Evidence of use of both sets of results from Ben and Fred.

Process of dividing by 100.

P(heads) = 0.48 or equivalent

P(tails) = 0.52 or equivalent

Learning checklist

You have now learned how to:

• Use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size

The next lessons are

• How to calculate probability
• Combined events probability
• Describing probability

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Theoretical Probability & Experimental Probability

Related Pages Probability Tree Diagrams Probability Without Replacement Probability Word Problems More Lessons On Probability

In these lessons, we will look into experimental probability and theoretical probability.

The following table highlights the difference between Experimental Probability and Theoretical Probability. Scroll down the page for more examples and solutions.

Printable Probability (Equally Likely Outcomes) Probability (Not Equally Likely Outcomes) Probability Tree Diagrams

Online Probability Problems Complementary Probability Probability Problems Probability & Geometry Mutually Exclusive Probability Independent Events Probability Dependent Events Probability

How To Find The Experimental Probability Of An Event?

Step 1: Conduct an experiment and record the number of times the event occurs and the number of times the activity is performed.

Step 2: Divide the two numbers to obtain the Experimental Probability.

How To Find The Theoretical Probability Of An Event?

The Theoretical Probability of an event is the number of ways the event can occur (favorable outcomes) divided by the number of total outcomes.

What Is The Theoretical Probability Formula?

The formula for theoretical probability of an event is

Experimental Probability

One way to find the probability of an event is to conduct an experiment.

Example: A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the experimental probability of getting a blue marble.

Solution: Take a marble from the bag. Record the color and return the marble. Repeat a few times (maybe 10 times). Count the number of times a blue marble was picked (Suppose it is 6).

How to find and use experimental probability?

The following video gives another example of experimental probability.

How the results of the experimental probability may approach the theoretical probability?

Example: The spinner below shows 10 equally sized slices. Heather spun 50 times and got the following results. a) From Heather’s’ results, compute the experimental probability of landing on yellow. b) Assuming that the spinner is fair, compute the theoretical probability of landing in yellow.

Theoretical Probability

We can also find the theoretical probability of an event.

Example: A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the theoretical probability of getting a blue marble.

Solution: There are 8 blue marbles. Therefore, the number of favorable outcomes = 8. There are a total of 20 marbles. Therefore, the number of total outcomes = 20

Example: Find the probability of rolling an even number when you roll a die containing the numbers 1-6. Express the probability as a fraction, decimal, ratio and percent.

Solution: The possible even numbers are 2, 4, 6. Number of favorable outcomes = 3. Total number of outcomes = 6

Comparing Theoretical And Experimental Probability

The following video gives an example of theoretical and experimental probability.

Example: According to theoretical probability, how many times can we expect to land on each color in a spinner, if we take 16 spins? Conduct the experiment to get the experimental probability.

We will then compare the Theoretical Probability and the Experimental Probability.

The following video shows another example of how to find the theoretical probability of an event.

A spinner is divided into eight equal sectors, numbered 1 through 8. a) What is the probability of spinning an odd numbers? b) What is the probability of spinning a number divisible by 4? b) What is the probability of spinning a number less than 3?

A spinner is divided into eight equal sectors, numbered 1 through 8. a) What is the probability of spinning a 2? b) What is the probability of spinning a number from 1 to 4? b) What is the probability of spinning a number divisible by 2?

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Experimental Probability – Formula, Definition With Examples

Updated on January 12, 2024

Welcome to another exciting journey with us at Brighterly , where we make the learning of complex mathematical concepts a fun and engaging process. Today, we embark on a venture into the world of experimental probability, a vital aspect of mathematics that breathes life into numbers through practical, real-world experiences. But what exactly is experimental probability, and how does it differ from theoretical probability? How can we calculate experimental probability, and how is it applicable in our everyday lives? This article aims to answer these questions and more, unraveling the mysteries of experimental probability in an easy-to-understand and approachable manner.

Our trip into the world of experimental probability will cover the core concepts, definitions, and the all-important formula that underpins this fascinating area of mathematics. We’ll take a step back to appreciate the broader context of probability before focusing our lens on experimental probability, understanding its properties and contrasts with theoretical probability. With plenty of examples and practice problems, you’ll have a firm grasp on experimental probability, ready to see and use it in the world around you!

What Is Experimental Probability?

Probability, as a field of mathematics, often focuses on predicting the likelihood of certain events. However, it’s important to note that there are two main types of probability: theoretical and experimental. In this article, we will zero in on experimental probability.

Experimental probability, also known as empirical probability, is all about actual experiments and real-world observations. The main idea behind experimental probability is that it calculates the chances of an event happening based on the actual results of an experiment. This method of calculation is particularly interesting because it revolves around practical events that have already occurred, rather than theoretical or hypothetical situations.

In experimental probability, we conduct a certain experiment multiple times and observe the number of times a specific event occurs. This might sound quite complex, but we’ll dive into this concept with a greater depth in the upcoming sections, making it easily understandable.

Definition of Probability

Before we delve into experimental probability, let’s take a step back and understand the basic concept of probability. Probability is defined as a branch of mathematics that measures the likelihood of events to occur. It’s expressed as a number between 0 and 1, where 0 indicates impossibility and 1 signifies certainty.

For example, consider flipping a fair coin. The probability of landing a “heads” is 1 out of 2, or 0.5, meaning there’s a 50% chance to get a “heads”. The same applies to “tails”.

Understanding probability can help us make predictions about the outcomes of a random event and aids in making informed decisions in various aspects of life including gaming, statistics, and even weather forecasting.

Definition of Experimental Probability

Moving on to experimental probability, it is defined as the ratio of the number of times an event occurs to the total number of trials or times the activity is performed. The experimental probability of an event is calculated by conducting an experiment and recording the results.

For instance, let’s say we roll a dice 100 times, and the number “4” comes up 15 times. Here, the experimental probability of rolling a “4” would be the number of successful outcomes (rolling a “4”) divided by the total number of outcomes (total dice rolls), or 15/100 = 0.15.

In other words, experimental probability is the actual probability obtained from the direct observation or testing during an experiment. Unlike theoretical probability, it doesn’t rely on the inherent nature of the experiment, but rather on the actual data collected.

Properties of Probability

Understanding the properties of probability can provide us with insights about how probability functions. Here are some of the essential properties:

• The probability of an event ranges from 0 to 1.
• The sum of probabilities of all possible outcomes is always 1.
• The probability of the complement of an event (an event not happening) is 1 minus the probability of the event.
• If two events are mutually exclusive (they can’t occur at the same time), the probability of either event occurring is the sum of their individual probabilities.

These properties provide a foundational understanding of how probability works, whether it’s theoretical or experimental probability.

Properties of Experimental Probability

The properties of experimental probability are closely tied to those of theoretical probability, but with an emphasis on the data collected through experimentation. Here are the primary properties:

• Experimental probability also ranges from 0 to 1. An experimental probability of 0 means the event never happened in the experiment, and a probability of 1 means the event always occurred.
• As more trials are conducted, the experimental probability tends to approach the theoretical probability, given that the experiment is unbiased. This is known as the law of large numbers.
• Like in theoretical probability, the sum of experimental probabilities of all possible outcomes is 1.

Understanding these properties can greatly aid in interpreting the results of experiments and the likelihood of outcomes.

Difference Between Theoretical and Experimental Probability

The primary difference between theoretical and experimental probability lies in their calculation and interpretation. Theoretical probability is based on the possible outcomes in theory. It assumes that all outcomes are equally likely, which isn’t always the case in real-world scenarios.

On the other hand, experimental probability is based on actual data collected from performed experiments. It deals with the frequency of occurrence of an event, providing a more empirical perspective on probability. For example, in theory, the probability of rolling a “6” on a fair die is 1/6. However, in an actual experiment of, say, 60 rolls, we might roll a “6” only 8 times. The experimental probability then becomes 8/60 or 0.1333.

Formula of Experimental Probability

The formula of experimental probability is quite straightforward:

By using this formula, we can calculate the experimental probability of an event based on the results of an actual experiment or observation.

Understanding the Formula of Experimental Probability

To understand the formula of experimental probability, let’s revisit the dice rolling example. If you roll a die 100 times, and the number “4” comes up 20 times, then the experimental probability of rolling a “4” is:

Experimental Probability = Number of times event occurs / Total number of trials

Experimental Probability = 20 / 100 = 0.2

Hence, based on the results of this experiment, the experimental probability of rolling a “4” is 0.2 or 20%.

This formula essentially calculates the frequency of occurrence of an event in an experiment, providing a realistic interpretation of probability.

Calculating Experimental Probability Using the Formula

Let’s consider another example to illustrate the calculation of experimental probability using the formula. Imagine you’re shooting basketball hoops. You take 30 shots and make 18 of them. What’s the experimental probability of making a shot?

Applying the formula, we get:

Experimental Probability = 18 / 30 = 0.6

So, the experimental probability of making a shot, based on this experiment, is 0.6 or 60%.

Practice Problems on Experimental Probability

To better understand how to calculate experimental probability, let’s work through some practice problems:

• A spinner with 8 equal sections numbered 1 to 8 is spun 50 times. The number 3 comes up 7 times. What is the experimental probability of the spinner landing on 3?
• In a school, a survey of what pet each student has at home is conducted. Out of 200 students, 45 have dogs. What is the experimental probability that a randomly selected student has a dog?
• In a bag of 100 marbles, 25 are red, and the rest are blue. If you randomly select a marble, replace it, and repeat this 100 times, and you get a red marble 28 times, what is the experimental probability of drawing a red marble?
• Experimental Probability = 7 / 50 = 0.14
• Experimental Probability = 45 / 200 = 0.225
• Experimental Probability = 28 / 100 = 0.28

And that wraps up our enlightening exploration of experimental probability! With Brighterly, we’ve unpacked this fascinating mathematical concept, revealing its significance and wide-ranging applications in our everyday life. Experimental probability, with its basis in real-world observations, lends us the power to anticipate outcomes based on our experiences, paving the way for more informed decision-making.

From understanding the basic definition of probability to distinguishing between theoretical and experimental probability and mastering the formula of experimental probability, we hope you’re now well-equipped to navigate the captivating world of probability. Remember, probability isn’t just a concept confined within the pages of a mathematics textbook; it’s very much a part of the world around us, informing everything from weather forecasts to game strategy and risk analysis.

So, the next time you play a game of cards, shoot hoops, or even make a decision based on certain outcomes, remember the role of experimental probability! As always, the team at Brighterly is dedicated to making the learning of complex concepts enjoyable, ensuring you have fun on your journey of exploration. Stay tuned for more exciting math adventures!

Frequently Asked Questions on Experimental Probability

What is experimental probability.

Experimental probability is a probability value that is based on actual experiments or observations. In other words, it’s a type of probability that quantifies the ratio of the number of times an event occurs to the total number of trials or times an activity is performed. For example, if you flip a coin 100 times and it lands on heads 45 times, the experimental probability of getting heads is 45/100 = 0.45 or 45%.

How do you calculate experimental probability?

Calculating experimental probability is straightforward. It involves dividing the number of times an event occurs by the total number of trials. For instance, if you roll a die 60 times and get a ‘6’ on 10 occasions, the experimental probability of rolling a ‘6’ would be 10 (number of successful outcomes) divided by 60 (total number of outcomes), which equals 0.1667 or 16.67%.

What is the difference between theoretical and experimental probability?

Theoretical probability and experimental probability differ in their calculation and interpretation. Theoretical probability is a type of probability that assumes that all outcomes of an experiment are equally likely. It’s calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

On the other hand, experimental probability doesn’t rely on the assumption of equally likely outcomes but instead depends on actual data collected from conducted experiments. It deals with the frequency or proportion of times an event occurs based on experimental data.

Why is experimental probability important?

Experimental probability plays a crucial role in various fields and everyday life. Its importance lies in its basis on real-world data, which makes it a practical tool for predicting the likelihood of outcomes based on past experiences. Experimental probability is utilized in various sectors such as statistics, data analysis, gaming, weather forecasting, and in the medical field, among others. It also plays a key role in empirical research, where it aids in providing evidence-based conclusions.

• Wikipedia – Probability
• NCBI – Probability in Health
• Gov.uk – Understanding Uncertainty and Risk

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.

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What is experimental probability? 

Practice questions, experimental probability – explanation & examples.

Experimental probability is the probability determined based on the results from performing the particular experiment. 

In this lesson we will go through:

• The meaning of experimental probability
• How to find experimental probability

The ratio of the number of outcomes favorable to an event to the total number of trials of the experiment.

Experimental Probability can be expressed mathematically as: 

$P(\text{E}) = \frac{\text{number of outcomes favorable to an event}}{\text{total number of trials of the experiment}}$

Let’s go back to the die tossing example. If after 12 throws you get one 6, then the experimental probability is $\frac{1}{12}$.  You can compare that to the theoretical probability. The theoretical probability of getting a 6 is $\frac{1}{6}$. This means that in 12 throws we would have expected to get 6 twice. 

Similarly, if in those 12 tosses you got a 1 five times, the experimental probability is $\frac{5}{12}$. 

How do we find experimental probability?

Now that we understand what is meant by experimental probability, let’s go through how it is found. 

To find the experimental probability of an event, divide the number of observed outcomes favorable to the event by the total number of trials of the experiment. 

Let’s go through some examples. 

Example 1:  There are 20 students in a class. Each student simultaneously flipped one coin. 12 students got a Head. From this experiment, what was the experimental probability of getting a head?

Number of coins showing Heads: 12

Total number of coins flipped: 20

$P(\text{Head}) = \frac{12}{20} = \frac{3}{5}$ 

Example 2:  The tally chart below shows the number of times a number was shown on the face of a tossed die. 

a. What was the probability of a 3 in this experiment?

b. What was the probability of a prime number?

First, sum the numbers in the frequency column to see that the experiment was performed 30 times. Then find the probabilities of the specified events. 

a. Number of times 3 showed = 7

Number of tosses = 30

$P(\text{3}) = \frac{7}{30}$ 

b. Frequency of primes = 6 + 7 + 2 = 15

Number of trials = 30 

$P(\text{prime}) = \frac{15}{30} = \frac{1}{2}$

Experimental probability can be used to predict the outcomes of experiments. This is shown in the following examples. 

Example 3: The table shows the attendance schedule of an employee for the month of May.

a. What is the probability that the employee is absent? 

b. How many times would we expect the employee to be present in June?

a. The employee was absent three times and the number of days in this experiment was 31. Therefore:

$P(\text{Absent}) = \frac{3}{31}$

b.  We expect the employee to be absent

$\frac{3}{31} × 30 = 2.9 ≈ 3$ days in June 

Example 4:  Tommy observed the color of cars owned by people in his small hometown. Of the 500 cars in town, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were grey. 

a. What is the probability that a car is red?

b. If a new car is bought by someone in town, what color do you think it would be? Explain. 

a. Number of red cars = 50 

Total number of cars = 500 

$P(\text{red car}) = \frac{50}{500} = \frac{1}{10}$ 

b. Based on the information provided, it is most likely that the new car will be black. This is because it has the highest frequency and the highest experimental probability. 

Now it is time for you to try these examples. 

The table below shows the colors of jeans in a clothing store and their respective frequencies. Use the table to answer the questions that follow.

• What is the probability of selecting a brown jeans?
• What is the probability of selecting a blue or a white jeans?

On a given day, a fast food restaurant notices that it sold 110 beef burgers, 60 chicken sandwiches, and 30 turkey sandwiches. From this observation, what is the experimental probability that a customer buys a beef burger?

Over a span of 20 seasons, a talent competition notices the following. Singers won 12 seasons, dancers won 2 seasons, comedians won 3 seasons, a poet won 1 season, and daring acts won the other 2 seasons. 

a. What is the experimental probability of a comedian winning  a season?

b. From the next 10 seasons, how many winners do you expect to be dancers?

Try this at home! Flip a coin 10 times. Record the number of tails you get. What is your P(tail)?

Number of brown jeans = 25

Total Number of jeans = 125

$P(\text{brown}) = \frac{25}{125}  = \frac{1}{5}$

Number of jeans that are blue or white = 75 + 20 = 95

$P(\text{blue or white}) = \frac{95}{125} = \frac{19}{25}$

Number of beef burgers = 110 

Number of burgers (or sandwiches) sold = 200 

$P(\text{beef burger}) = \frac{110}{200} = \frac{11}{20}$ 

a. Number of comedian winners = 3

Number of seasons = 20 

$P(\text{comedian}) = \frac{3}{20}$ 

b. First find the experimental probability that the winner is a dancer. 

Number of winners that are dancers = 2 

$P(\text{dancer}) = \frac{2}{20} = \frac{1}{10}$ 

Therefore we expect 

$\frac{1}{10} × 10 = 1$ winner to be a dancer in the next 10 seasons.

To find your P(tail) in 10 trials, complete the following with the number of tails you got. 

$P(\text{tail}) = \frac{\text{number of tails}}{10}$ 

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Question: If you roll a die six times and get an even number two times, what is the experimental probability of rolling an even number?*answers are fractions

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If a fair die is rolled 3 times, what are the odds of getting an even number on each of the first 2 rolls, and an odd number on the third roll?

I think the permutations formula is needed i.e. $n!/(n-r)!$ because order matters but I'm not sure if n is 3 or 6 and what would r be?

Any help would be much appreciated!

• probability
• permutations

5 Answers 5

Let's calculate the probability, then convert that to odds.

On a fair die, half the numbers are even and half the numbers are odd. So, the probability for a single roll of getting an even number or an odd number is $\dfrac{1}{2}$ .

The probability for a specific roll are unaffected by previous rolls, so we can apply the product principle and multiply probabilities for each roll. Each roll has probability of $\dfrac 1 2$ of obtaining the desired result. So, we have:

$$P(E,E,O) = \dfrac 1 2 \cdot \dfrac 1 2 \cdot \dfrac 1 2 = \dfrac 1 8$$

Now, the probability of that not happening is $$1-\dfrac 1 8 = \dfrac 7 8$$

So, the odds are 7:1 against the desired outcome.

I think you might be over complicating things.

It has to be even on the first two, the probability of this is $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}.$

The probability of odd on the third roll is also $\frac{1}{2}$ so your final probability is $\frac{1}{8}.$

• $\begingroup$ This is simpler. $\endgroup$ –  Snowcrash Commented Sep 21, 2018 at 14:44
• $\begingroup$ ... this is using odds... not probability $\endgroup$ –  Jason Kim Commented Sep 22, 2018 at 3:36
• $\begingroup$ @jason Kim what do you think the distinction is..? $\endgroup$ –  MRobinson Commented Sep 22, 2018 at 7:59
• $\begingroup$ Odds for is (prob)/(1-prob) which demonstrates the difference... $\endgroup$ –  Jason Kim Commented Sep 24, 2018 at 0:58
• $\begingroup$ @JasonKim if the probability is $\frac{1}{8}$ then the odds are $1$ in $8$. If you want to write the odds like a betting shop then you'll have $1:7$. I struggle to see how you believe I have used odds and not probability. $\endgroup$ –  MRobinson Commented Sep 24, 2018 at 7:32

Given a fair die, the probability of any defined sequence of three evens or odds is the same, namely, $1/2 \times 1/2 \times 1/2 = 1/8$ . So the odds are $7$ to $1$ against.

Assuming the die is just marked even and odd rather than with numbers, there are eight orderings. They range from all odd to all even: OOO, OOE, OEO, EOO, OEE, EOE, EEO, EEE. In your formula:

$$\frac{n!}{(n-r)!}$$

You are missing that you don't care about the orders of the duplicates. So you need

$$\frac{n!}{(n-r)!r!}$$

The $n$ is the total number of rolls, the $r$ is the number of either evens or odds (it's symmetric). But to get the total number of orderings, you need to add these:

$$\sum\limits_{r=0}^n\frac{n!}{(n-r)!r!}$$

Now substitute 3 for $n$ .

$$\sum\limits_{r=0}^3\frac{3!}{(3-r)!r!}$$

Unrolling that, we get

$$\frac{3!}{3!0!} + \frac{3!}{2!1!} + \frac{3!}{1!2!} + \frac{3!}{0!3!}$$

$$1 + 3 + 3 + 1$$

So we have one all odds, three with one even, three with two evens, and one all even. That's eight total.

Another way of thinking of this is that there is only one ordering of all odd numbers or all even numbers while there are three places where the lone odd or even number can be.

Of those eight, how many fit your parameters? Exactly one, OOE. So one in eight or $\frac{1}{8}$ .

As others have already noted, you could get that much more easily by simply figuring that you have a one in two chance of getting the result you need for each roll. There's three rolls, so $(\frac{1}{2})^3 = \frac{1}{8}$ .

If you want to treat 1, 3, and 5 as different values and 2, 4, and 6 as different values, you can. But it is much easier to think of them as just odd or even. Because you don't want to try write this out for $6^3 = 216$ orderings. And in the end, you will get the same basic result. You will have twenty-seven OOE orderings, which is again one eighth of the total. This is because there are three possible values for each, 1, 3, and 5 for the two odds and 2, 4, and 6 for the even. And $\frac{27}{216} = \frac{1}{8}$ .

Permutations leads you down a harder path. It's easier to think just in terms of probability or even ordering.

All of the above or below :-} answers are correct that is 50:50 for each throw so your "formula" is

  n (2 -1):1 against where n is the number of throws

• 1 $\begingroup$ Welcome to MathSE. Please read this tutorial on how to typeset mathematics on this site. $\endgroup$ –  N. F. Taussig Commented Sep 21, 2018 at 22:08

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Math in Society: Mathematics for liberal arts majors

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Section 4.2 Theoretical Probability

• Write the sample space for theoretical probability situations
• Identify certain and impossible events
• Calculate the theoretical probability of a complement
• Determine the difference between empirical and theoretical probability
• Explain the Law of Large Numbers
• Identify independent and dependent events
• Calculate “and” theoretical probabilities
• Identify overlapping and disjoint sets
• Calculate “or” theoretical probabilities
• Calculate probability values for simple games

Subsection 4.2.1 Basic Probability Concepts

Subsection 4.2.2 experiment, subsection 4.2.3 events and outcomes, subsection 4.2.4 sample space, example 4.2.2 ., subsection 4.2.5 equally likely outcomes, theoretical probability., example 4.2.3 ..

Example 4.2.4 .

• rolling a 6.
• rolling a number that is at least 4.
• rolling an even number.
• rolling a 5 or a 3.
• There is only one way to roll a 6, so \(P(\text{rolling a 6}) =\frac{1}{6}\) or approximately 16.7% There is a 16.7% chance of rolling a 6.
• In probability we will often come across the phrases “at least” and “at most.” At least means that value or greater. At most means that value or less. Since we are looking for the probability of rolling a number that is at least 4, we need the number of outcomes that are 4 or greater. There are 3 values that meet this condition: 4, 5, and 6. The probability is \(P(\text{rolling a number that is at least 4})=\frac{3}{6}=\frac{1}{2}\) or 50%
• Half of the numbers on a die are even, so we calculate: \(P(\text{rolling an even number})=\frac{3}{6}=\frac{1}{2}\) or 50%
• There are two ways to roll a 5 or a 3, so \(P(\text{rolling a 5 or a 3})=\frac{2}{6}=\frac{1}{3}\) or approxiamtely 33.3%. There is a 33.3% chance of rolling a 5 or a 3.

Example 4.2.5 .

Subsection 4.2.6 certain and impossible events, example 4.2.6 ..

• What is the probability of rolling an odd or even number on a six-sided die?
• What is the probability of rolling an 8 on a six-sided die?
• Since all the numbers are either even or odd, this event includes all of the outcomes in the sample space. This event is certain. \(P(\text{odd or even})=\frac{6}{6}=1\text{ or } 100\%\)
• Since 8 is not one of the outcomes in the sample space, the event is impossible. \(P(\text{roll an 8})=\frac{0}{8}=0\text{ or } 0\%\)

Subsection 4.2.7 Complementary Events

Complement of an Event.

Example 4.2.7 ., subsection 4.2.8 experimental vs. theoretical probability, example 4.2.8 ..

Subsection 4.2.9 The Law of Large Numbers

Subsection 4.2.10 probability of compound events, subsection 4.2.11 “and” probabilities, subsection 4.2.12 independent and dependent events, example 4.2.9 ..

• Flipping a fair coin twice and getting heads both times.
• Selecting a president and then a vice president at random from a pool of five equally qualified individuals.
• The event that it will rain in Portland tomorrow and the event that it will rain in Beaverton tomorrow.
• Wearing your lucky socks and getting an A on your exam.
• The probability of getting heads on the first flip is 0.5 or 50%. After flipping heads, the probability of getting heads on the second flip is still 0.5 or 50%. Since the probability of flipping heads on the second flip did not change because we flipped heads on the first flip, the events are independent .
• Since two different people will be put in the role of president and vice president, we are drawing without replacement and the events are therefore dependent .
• If it is raining in Portland it is more likely that it will rain in Beaverton, so the events are dependent .
• Although there may some sort of placebo effect at play in terms of confidence and persistence, the socks you wear do not have a direct effect on how well you do on your exam, so these events are independent .

Example 4.2.10 .

• two red Legos in a row if we put the first red Lego back in the bag?
• two red Legos in a row if we don’t put the first Lego back in the bag?
• a red Lego and then a green Lego if we do not put the red Lego back in the bag?
• Since the outcomes are equally likely, the probability of selecting a red Lego is the number of red Legos divided by the total number of Legos, or \(P(\text{red})=\frac{6}{13}\text{.}\) If we replace the red Lego we selected (selections are independent), we go back to having 6 red Legos in the bag of 13 Legos total. Therefore, \begin{align*} P(\text{red and then red})\amp=P(\text{red})\cdot P(\text{red})\\ \amp=\frac{6}{13}\cdot\frac{6}{13}\\ \amp\approx 0.213 \text{ or } 21.3\% \end{align*}
• If we do not replace the first red Lego (selections are dependent), then on our second draw there will only be 5 red Legos remaining, and 12 Legos in total. Therefore, \begin{align*} P(\text{red and then red})\amp=P(\text{red})\cdot P(\text{red given red taken out})\\ \amp=\frac{6}{13}\cdot\frac{5}{12}\\ \amp\approx 0.192 \text{ or } 19.2\% \end{align*}
• The probability of selecting a red Lego on the first draw is the same as in parts a and b. Since we are not putting the red Lego back into the bag, we will have only 12 Lego left in total, of which 4 are green. Therefore, \begin{align*} P(\text{red and then green})\amp=P(\text{red})\cdot P(\text{green given red taken out})\\ \amp=\frac{6}{13}\cdot\frac{4}{12}\\ \amp=\frac{6}{13}\cdot\frac{1}{3}\\ \amp\approx 0.154 \text{ or } 15.4\% \end{align*}

Example 4.2.11 .

Subsection 4.2.13 “or” probabilities, subsection 4.2.14 overlapping or disjoint sets.

Example 4.2.12 .

• getting a yellow prize.
• getting a red or yellow prize.
• getting a prize that is yellow or an eraser.
• Since yellow is a single event, we just need to know how many prizes there are in total, and how many of the prizes are yellow. The yellow prizes include the 10 yellow erasers and the 8 yellow pencil sharpeners. \begin{align*} P(\text{yellow})\amp=\frac{18}{33}\\ \amp\approx 0.545 \text{ or } 54.5\% \end{align*}
• For a red or yellow prize, the set of red and the set of yellow do not overlap. They are disjoint sets, so we will add the probability of getting a red prize to the probability of getting a yellow prize. \begin{align*} P(\text{red or yellow})\amp=P(\text{red})+P(\text{yellow})\\ \amp=\frac{9}{33}+\frac{18}{33}\\ \amp=\frac{27}{33}\\ \amp\approx 0.818 \text{ or } 81.8\% \end{align*}
• To find the probability of getting a prize that is yellow or an eraser, we need to be careful because these are overlapping sets. There are two ways to calculate this, and it is a lot like what we did with contingency tables. The first way is to add all the items separately, being careful not to double count. \begin{align*} P(\text{yellow or eraser})\amp=P(\text{yellow eraser})+P(\text{yellow pencil sharpener})+P(\text{green eraser})\\ \amp=\frac{10}{33}+\frac{8}{33}+\frac{6}{33}\\ \amp=\frac{24}{33}\\ \amp\approx 0.727 \text{ or } 72.7\% \end{align*} The second way is to count the total of yellow items and the total of erasers, but the yellow erasers are in both sets, or the overlap. We would be counting them twice and so we subtract their joint probability. \begin{align*} P(\text{yellow or eraser})\amp=P(\text{yellow})+P(\text{eraser})-P(\text{yellow and eraser})\\ \amp=\frac{18}{33}+\frac{16}{33}-\frac{10}{33}\\ \amp=\frac{24}{33}\\ \amp\approx 0.727 \text{ or } 72.7\% \end{align*}

“Or” Probabilities.

Example 4.2.13 ..

Exercises 4.2.15 Exercises

• Drawing a white marble.
• Drawing a red marble.
• Drawing a green marble.
• Drawing two yellow marbles if you draw with replacement.
• Drawing first a red marble then a white marble if marbles are drawn without replacement.
• an even number.
• a number less than 3.
• a number other than 8.
• a 6 on both rolls?
• a 5 on the first roll and an even number on the second roll?
• What is probability the minutes reading is 15?
• What is the probability the minutes reading is 15 or less?
• and getting a head each time?
• not getting a head at all?
• and getting a sum greater than or equal to 7?
• getting an even sum or a sum greater than 7?
• both pieces are dotted?
• the first piece is black, and the second piece is dotted?
• one piece is black, and one piece is striped?
• a 3 on all five dice.
• at least one of the die shows a 3.
• a spade or a club?
• a diamond or a 5?

Probability Calculator

Table of contents

With the probability calculator, you can investigate the relationships of likelihood between two separate events . For example, if the chance of A happening is 50%, and the same for B, what are the chances of both happening, only one happening, at least one happening, or neither happening, and so on.

Our probability calculator gives you six scenarios, plus 6 more when you enter in how many times the "die is cast", so to speak. As long as you know how to find the probability of individual events, it will save you a lot of time.

Reading on below, you'll:

• Discover how to use the probability calculator properly;
• Check how to find the probability of single events;
• Read about multiple examples of probability usage, including conditional probability formulas;
• Study the difference between a theoretical and empirical probability; and
• Increase your knowledge about the relationship between probability and statistics.

Did you come here specifically to check your odds of winning a bet or hitting the jackpot? Our odds calculator and lottery calculator will assist you!

How to find the probability of events? – probability definition

The basic definition of probability is the ratio of all favorable results to the number of all possible outcomes.

Allowed values of a single probability vary from 0 to 1 , so it's also convenient to write probabilities as percentages. The probability of a single event can be expressed as such:

• The probability of A : P(A) ,
• The probability of B : P(B) ,
• The probability of + : P(+) ,
• The probability of ♥ : P(♥) , etc.

Let's take a look at an example with multi-colored balls. We have a bag filled with orange, green, and yellow balls. Our event A is picking a random ball out of the bag . We can define Ω as a complete set of balls. The probability of event Ω , which means picking any ball, is naturally 1. In fact, a sum of all possible events in a given set is always equal to 1 .

Now let's look at something more challenging – what's the likelihood of picking an orange ball? To answer this question, you have to find the number of all orange marbles and divide it by the number of all balls in the bag. You can do it for any color, e.g., yellow, and you'll undoubtedly notice that the more balls in a particular color, the higher the probability of picking it out of the bag if the process is totally random.

Check out our probability calculator 3 events and conditional probability calculator for determining the chances of multiple events.

We can define a complementary event , written as Ā or A' , which means not A . In our example, the probability of picking out NOT an orange ball is evaluated as a number of all non-orange ones divided by all marbles. The sum P(A) + P(Ā) is always 1 because there is no other option like half of a ball or a semi-orange one.

Now, try to find the probability of getting a blue ball. No matter how hard you try, you will fail because there is not even one in the bag, so the result is equal to 0.

We use intuitive calculations of probability all the time. Knowing how to quantify likelihood is essential for statistical analysis. It allows you to measure this otherwise nebulous concept called "probability". Furthermore, given a discrete dataset, the relative frequency for each value is synonymous with the probability of their occurrence.

Are you looking for something slightly different? Take a look at our post-test probability calculator . 🎲

How to use the probability calculator?

To make the most of our calculator, you'll need to take the following steps:

1. Define the problem you want to solve.

Your problem needs to be condensed into two independent events.

2. Find the probability of each event.

Now, when you know how to estimate the likelihood of a single event, you only need to perform the task and obtain all of the necessary values.

3. Type the percentage probability of each event in the corresponding fields.

Once they're in, the probability calculator will immediately populate with the exact likelihood of 6 different scenarios:

• Both events will happen;
• At least one of the events will happen;
• Exactly one of the events will happen;
• Neither of the events will happen;
• Only the first event won't happen; and
• Only the second event won't happen

You can also choose to see all of the above. Additionally, the calculator can also show the probability of six more scenarios, given a certain number of trials:

• A always occurring;
• A never occurring;
• A occurring at least once;
• B always occurring;
• B never occurring; and
• B occurring at least once.

You can change the number of trials and any other field in the calculator, and the other fields will automatically adjust themselves. This feature saves a ton of time if you want to find out, for example, what the probability of event B would need to become in order to make the likelihood of both occurring 50%.

If the set of possible choices is extremely large and only a few outcomes are successful, the resulting probability is tiny, like P(A) = 0.0001 . It's convenient to use scientific notation in order not to mix up the number of zeros.

Conditional probability

One of the most crucial considerations in the world of probabilities is whether the events are dependent or not. Two events are independent if the occurrence of the first one doesn't affect the likelihood of the occurrence of the second one . For example, if we roll a perfectly balanced standard cubic die, the possibility of getting a two ⚁ is equal to 1/6 (the same as getting a four ⚃ or any other number).

Let's say you have two dice rolls, and you get a five ⚄ in the first one. If you ask yourself what's the probability of getting a two ⚁ in the second turn, the answer is 1/6 once again because of the independence of events.

The way of thinking, as well as calculations, change if one of the events interrupts the whole system. This time we're talking about conditional probability .

Let's say we have 10 different numbered billiard balls, from ➀ to ➉. You choose a random ball, so the probability of getting the ➆ is precisely 1/10 . Suppose you picked the three ➂ and removed it from the game . Then you ask yourself, once again, what is the chance of getting the seven ➆. The situation changed because there is one ball with ➆ out of nine possibilities, which means that the probability is 1/9 now. In other words, the question can be asked: "What's the probability of picking ➆, IF the first ball was ➂?"

Let's look at another example: imagine that you are going to sit an exam in statistics. You know from your older colleagues that it's challenging, and the probability that you pass in the first term is 0.5 ( 18 out of 36 students passed last year). Then let's ask yourself a question: "What's the probability of passing IF you've already studied the topic?" 20 people admitted to reviewing their notes at least once before the exam, and 16 out of those succeeded, which means that the answer to the last question is 0.8 . This result indicates that this additional condition really matters if we want to find whether studying changes anything or not.

If you still don't feel the concept of conditional probability, let's try with another example: you have to drive from city X to city Y by car. The distance between them is about 150 miles. On the full tank, you can usually go up to 400 miles. If you don't know the fuel level, you can estimate the likelihood of successfully reaching the destination without refueling. And what if somebody has already filled the tank? Now you're almost sure that you can make it unless other issues prevent it.

Conditional probability formula

The formal expression of conditional probability, which can be denoted as P(A|B) , P(A/B) or P B (A) , can be calculated as:

P(A|B) = P(A∩B) / P(B) ,

where P(B) is the probability of an event B , and P(A∩B) is the joint of both events. On the other hand, we can estimate the intersection of two events if we know one of the conditional probabilities:

• P(A∩B) = P(A|B) * P(B) or
• P(A∩B) = P(B|A) * P(A) .

It's better to understand the concept of conditional probability formula with tree diagrams. We ask students in a class if they like Math and Physics. An event M denotes the percentage that enjoys Math, and P the same for Physics:

There is a famous theorem that connects conditional probabilities of two events. It's named Bayes' theorem , and the formula is as follows:

P(A|B) = P(B|A) * P(A) / P(B)

You can ask a question: "What is the probability of A given B if I know the likelihood of B given A ?". This theorem sometimes provides surprising and unintuitive results. The most commonly described examples are drug testing and illness detection, which has a lot in common with the relative risk of disease in the population. Let's stick to the second one. In a group of 1000 people, 10 of them have a rare disease. Everybody had a test, which shows the actual result in 95% of cases. So now we want to find the probability of a person being ill if their test result is positive.

Without thinking, you may predict, by intuition, that the result should be around 90% , right? Let's make some calculations and estimate the correct answer.

• We will use a notation: H – healthy, I – ill, + – test positive, - – test negative.
• Rewrite information from the text above in a way of probabilities: P(H) = 0.99 , P(I) = 0.01 , P(+|I) = 0.95 , P(-|I) = 0.05 , P(+|H) = 0.05 , P(-|H) = 0.95 .
• Work out the total probability of a test to be positive: P(+) = P(+|I) * P(I) + P(+|H) * P(H) = 0.95 * 0.01 + 0.05 * 0.99 = 0.059 .
• Use the Bayes' theorem to find the conditional probability P(I|+) = P(+|I) * P(I) / P(+) = 0.95 * 0.01 / 0.059 = 0.161 .

Hmm... it isn't that high, is it? It turns out that this kind of paradox appears if there is a significant imbalance between the number of healthy and ill people , or in general, between two distinct groups. If the result is positive, it's always worth repeating the test to make an appropriate diagnosis.

Probability distribution and cumulative distribution function

We can distinguish between two kinds of probability distributions, depending on whether the random variables are discrete or continuous.

A discrete probability distribution describes the likelihood of the occurrence of countable, distinct events. One of the examples is binomial probability, which takes into account the probability of some kind of success in multiple turns, e.g., while tossing a coin. In contrast, in the Pascal distribution (also known as negative binomial) the fixed number of successes is given, and you want to estimate the total number of trials.

The Poisson distribution is another discrete probability distribution and is actually a particular case of binomial one, which you can calculate with our Poisson distribution calculator . The probability mass function can be interpreted as another definition of discrete probability distribution – it assigns a given value to any separate number. The geometric distribution is an excellent example of using the probability mass function.

A continuous probability distribution holds information about uncountable events. It's impossible to predict the likelihood of a single event (like in a discrete one), but rather that we can find the event in some range of variables. The normal distribution is one of the best-known continuous distribution. It describes a bunch of properties within any population, e.g., the height of adult people or the IQ dissemination. The function that describes the probability of seeing a result from a given range of values is called the probability density function .

If you are more advanced in probability theory and calculations, you definitely have to deal with SMp(x) distribution , which takes into account the combination of several discrete and continuous probability functions.

For each probability distribution, we can construct the cumulative distribution function (CDF) . It tells you what the probability is that some variable will take the value less than or equal to a given number .

Let's say you participate in a general knowledge quiz. The competition consists of 100 questions, and you earn 1 point for a correct answer, whereas for the wrong one, there are no points. Many people have already finished, and out of the results, we can obtain a probability distribution. Rules state that only 20% best participants receive awards, so you wonder how well you should score to be one of the winners. If you look at the graph, you can divide it so that 80% of the area below is on the left side and 20% of the results are on the right of the desired score. What you are actually looking for is a left-tailed p-value.

However, there is also another way to find it if we use a cumulative distribution function – just find the value 80% on the axis of abscissa and the corresponding number of points without calculating anything!

Theoretical vs experimental probability

Almost every example described above takes into account the theoretical probability. So a question arises: what's the difference between theoretical and experimental (also known as empirical) probability? The formal definition of theoretical probability is the ratio between the number of favorable outcomes to the number of every possible outcome . It relies on the given information, logical reasoning and tells us what we should expect from an experiment .

Just look at bags with colorful balls once again. There are 42 marbles in total, and 18 of them are orange. The game consists of picking a random ball from the bag and putting it back, so there are always 42 balls inside. Applying the probability definition, we can quickly estimate it as 18/42 , or simplifying the fraction, 3/7 . It means that if we pick 14 balls, there should be 6 orange ones.

On the other hand, the experimental probability tells us precisely what happened when we perform an experiment instead of what should happen. It is based on the ratio of the number of successful and the number of all trials . Let's stick with the same example – pick a random marble from the bag and repeat the procedure 13 more times. Suppose you get 8 orange balls in 14 trials. This result means that the empirical probability is 8/14 or 4/7 .

As you can see, your outcome differs from the theoretical one. It's nothing strange because when you try to reiterate this game over and over, sometimes, you will pick more, and sometimes you will get less, and sometimes you will pick exactly the number predicted theoretically. If you sum up all results, you should notice that the overall probability gets closer and closer to the theoretical probability . If not, then we can suspect that picking a ball from the bag isn't entirely random, e.g., the balls of different colors have unequal sizes, so you can distinguish them without having to look.

Probability and statistics

Both statistics and probability are the branches of mathematics and deal with the relationship of the occurrence of events . However, everyone should be aware of the differences which make them two distinct areas.

Probability is generally a theoretical field of math, and it investigates the consequences of mathematical definitions and theorems . In contrast, statistics is usually a practical application of mathematics in everyday situations and tries to attribute sense and understanding of the observations in the real world .

Probability predicts the possibility of events to happen , whereas statistics is basically analyzing the frequency of the occurrence of past ones and creates a model based on the acquired knowledge .

Imagine a probabilist playing a card game, which relies on choosing a random card from the whole deck, knowing that only spades win with predefined odds ratio. Assuming that the deck is complete and the choice is entirely random and equitable, they deduce that the probability is equal to ¼ and can make a bet.

A statistician is going to observe the game for a while first to check if, in fact, the game is fair. After verifying (with acceptable approximation) that the game is worth playing, then he will ask the probabilist what he should do to win the most.

Statistics within a large group of people – probability sampling

You've undoubtedly seen some election preference polls, and you may have wondered how they may be quite so precise in comparison to final scores, even if the number of people asked is way lower than the total population – this is the time when probability sampling takes place .

The underlying assumption, which is the basic idea of sampling, is that the volunteers are chosen randomly with a previously defined probability. We can distinguish between multiple kinds of sampling methods:

• Simple random sampling
• Cluster random sampling
• Systematic sampling
• Probability-proportional-to-size sampling
• Stratified random sampling
• Minimax sampling
• Accidental sampling
• Quota sampling
• Voluntary sampling
• Panel sampling
• Snowball sampling
• Line-intercept sampling
• Theoretical sampling

Each of these methods has its advantages and drawbacks, but most of them are satisfactory. Significant benefits of probability sampling are time-saving, and cost-effectiveness since a limited number of people needs to be surveyed. The simplicity of this procedure doesn't require any expertise and can be performed without any thorough preparation.

Practical application of probability theory

As you could have already realized, there are a lot of areas where the theory of probability is applicable. Most of them are games with a high random factor, like rolling dice or picking one colored ball out of 10 different colors, or many card games. Lotteries and gambling are the kinds of games that extensively use the concept of probability and the general lack of knowledge about it. Of course, somebody wins from time to time, but the likelihood that the person will be you is extremely small .

Probability theory is also used in many different types of problems. Especially when talking about investments, it is also worth considering the risk to choose the most appropriate option.

Our White Christmas calculator uses historical data and probability knowledge to predict the occurrence of snow cover for many cities during Christmas.

How do I calculate the probability of A and B?

If A and B are independent events , then you can multiply their probabilities together to get the probability of both A and B happening. For example, if the probability of A is 20% (0.2) and the probability of B is 30% (0.3) , the probability of both happening is 0.2 × 0.3 = 0.06 = 6% .

How do I calculate conditional probability?

To compute the conditional probability of A under B :

• Determine the probability of B , i.e., P(B) .
• Determine the probability of A and B , i.e., P(A∩B) .
• Divide the result from Step 2 by that of Step 1.
• That's it! The formula reads: P(A|B) = P(A∩B) / P(B) .

What's the probability of rolling 2 sixes?

If you are using fair dice, the probability of rolling two sixes will be 1/6 × 1/6 = 1/36 = 0.027 = 2.7% . That means it takes 36 dice rolls to expect rolling 2 sixes at least once, though there's no guarantee when it comes to probability.

How do I convert odds to percentage?

Convert the odds to a decimal number, then multiply by 100. For example, if the odds are 1 in 9, that's 1/9 = 0.1111 in decimal form. Then multiply by 100 to get 11.11% .

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ol{padding-top:0;}.css-63uqft ul:not(:first-child),.css-63uqft ol:not(:first-child){padding-top:4px;} Probabilities of single events

Probability of A: P(A)

Probability of B: P(B)

Which probability do you want to see?

Type of probability

Probability of the intersection of A and B.

Probabilities for a series of events

The probability of

...when trying

If you want to find the conditional probability, check out our Bayes' Theorem Calculator !