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Probability Distribution
Here we will learn about probability distribution, including theoretical probability, sample space, relative frequency, experimental probability and expected frequency.
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 a probability distribution?
A probability distribution is a summary of the probabilities of all possible outcomes of an experiment or situation, known as a random variable.
In GCSE mathematics they are usually described using a table.
For example,
This table shows the theoretical probability distribution for a fair katex is not defined-sided spinner.
As the spinner is fair, the probability of landing on each value should be the same.
Step-by-step guide: Theoretical probability
This table shows the probability distribution for a katex is not defined-sided spinner obtained after an experiment.
These probabilities are not all the same but still sum to katex is not defined
katex is not definedStep-by-step guide: Experimental probability
In a GCSE mathematics question, you may need to complete probability distribution tables or use the probabilities to solve problems and make predictions.
You are often asked to estimate the number of times a certain event will happen out of a given number of trials. This is known as expected frequency and can be worked out using this formula.
katex is not definedStep-by-step guide: Expected frequency
See also: Relative frequency
What is a probability distribution?
How to use probability distribution
In order to complete and use probability distribution:
- Identify the information provided to complete the distribution.
- Use the sum of the probabilities adding to katex is not defined or other information such as ratios or equations to find any missing probabilities.
- Use any relevant probabilities to answer any remaining questions or find expected frequencies.
Explain how to use probability distribution
Probability distribution worksheet
Get your free probability distribution worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREE Probability distribution worksheet
Get your free probability distribution worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEProbability distribution examples
Example 1: completing a probability distribution
A katex is not defined-sided spinner can land on blue, green or red.
The table shows the probabilities for the spinner landing on blue and green. Find the probability of the spinner landing on red.
- Identify the information provided to complete the distribution.
You have been given the probabilities of the spinner landing on blue and green, there is only the probability of the spinner landing on red missing.
2Use the sum of the probabilities adding to katex is not defined or other information such as ratios or equations to find any missing probabilities.
P(red) katex is not defined
3Use any relevant probabilities to answer any remaining questions or find expected frequencies.
No other information is required.
P(red) katex is not defined
Example 2: completing a probability distribution with more than one missing value
A bag only contains white, red and black counters. The probability of selecting a black counter is katex is not defined It is known that there are the same number of white counters as red counters.
Produce a table showing the probability distribution for the counters and find the expected frequency of red counters if there are known to be katex is not defined counters in total in the bag.
Identify the information provided to complete the distribution.
You have been given the probability of selecting a black counter and if the number of white counters is the same as the number of red counters, their probabilities will be the same.
Use the sum of the probabilities adding to katex is not defined or other information such as ratios or equations to find any missing probabilities.
P(white or red) katex is not defined
P(white) katex is not defined P(red), therefore each must be katex is not defined
This gives us the probability distribution for the counters.
Use any relevant probabilities to answer any remaining questions or find expected frequencies.
If there are katex is not defined counters in the bag, use the formula,
katex is not defined.
Number of red counters katex is not defined
Example 3: completing a probability distribution using a ratio or equation
A katex is not defined-sided spinner can land on blue, green, red or yellow.
The table shows the probabilities for the spinner landing on blue and green. The probability of the spinner landing on red is twice the probability of it landing on yellow.
Find the probability of the spinner landing on red or green.
Identify the information provided to complete the distribution.
You have been given the probabilities of the spinner landing on blue and green.
You also know P(red):P(yellow) katex is not defined:katex is not defined
Use the sum of the probabilities adding to katex is not defined or other information such as ratios or equations to find any missing probabilities.
P(red or yellow) katex is not defined
Share katex is not defined in the ratio katex is not defined:katex is not defined
katex is not defined
This gives P(red) katex is not defined and P(yellow) katex is not defined
You could have also formed an equation.
Let P(red) katex is not defined and P(yellow) katex is not defined
katex is not defined
Use any relevant probabilities to answer any remaining questions or find expected frequencies.
You can use the completed probability distribution.
P(red or green) katex is not defined P(red) katex is not defined P(green) katex is not defined
Common misconceptions
- Assuming that all probabilities are the same
It is possible for students to assume that all the probabilities in a distribution will be the same.
- Not knowing the meaning of the words “and“ and “or” in probability questions
It is important that students understand the difference between “and” and “or” and when they need to multiply probabilities and when they should add probabilities. In GCSE mathematics questions, the word “or” will mean they need to add the probabilities. “And” will involve multiplication.
Practice probability distribution questions
1. A bag contains only red, blue and green counters. Find the missing probability in this probability distribution.
2. A katex is not defined-sided spinner is numbered katex is not defined and katex is not defined
The probability of it landing on a katex is not defined is katex is not defined The probability of it landing on a katex is not defined is the same as landing on a katex is not defined
Find the probability of the spinner landing on a katex is not defined
3. A bag contains only blue, yellow and green counters.
The probability of selecting a blue counter is katex is not defined
The number of yellow counters is katex is not defined times the number of green counters.
Which is the correct probability distribution for the counters in the bag?
Share katex is not defined in the ratio katex is not defined:katex is not defined
4. A biased dice can land on a katex is not defined or katex is not defined
The table shows the probabilities of it landing on a katex is not defined or katex is not defined
The probabilities of it landing on a katex is not defined or katex is not defined are in the ratio katex is not defined:katex is not defined:katex is not defined Which is the correct probability distribution for the die?
katex is not defined Share katex is not defined in the ratio katex is not defined:katex is not defined:katex is not defined
katex is not defined
5. The table shows the probability distribution for a katex is not defined-sided spinner.
Find the value of katex is not defined
Form an equation and solve.
katex is not defined
6. A katex is not defined-sided spinner can land on blue, red, green or white.
The probabilities of landing on blue or green are shown in the table.
The spinner landing on white is twice as likely as it landing on red.
If the spinner is spun katex is not defined times, how many times would it be expected to land on white?
P(white) katex is not defined
katex is not defined
Probability distribution GCSE questions
1. A bag contains only white, yellow and red counters. There are twice as many red counters as yellow counters and three times as many yellow counters as white counters.
A counter is selected at random.
Fill in the table to show the probability distribution for the bag of counters.
(4 marks)
Process of forming a ratio or equation linking amounts of white, yellow and red counters. For example,
W:Y:R katex is not defined:katex is not defined:katex is not defined or equivalent.
(1)
P(white) katex is not defined or equivalent.
(1)
P(yellow) katex is not defined or equivalent.
(1)
P(red) katex is not defined or equivalent.
(1)
2. A spinning arrow can land on the numbers katex is not defined and katex is not defined
The probabilities of the arrow landing on katex is not defined or katex is not defined are shown in the table.
The arrow is half as likely to land on katex is not defined as it is katex is not defined
(a) Complete the missing probabilities in the table.
(b) The arrow is spun katex is not defined times. How many times would we expect it to land on katex is not defined
(5 marks)
(a)
Finding P(katex is not defined or katex is not defined) katex is not defined
(1)
Attempt to share katex is not defined in the ratio katex is not defined:katex is not defined or form an equation.
(1)
katex is not defined and katex is not defined shown.
(1)
(b)
Attempt to multiply katex is not defined by katex is not defined.
(1)
katex is not defined(1)
3. The table shows the probability of winning cash prizes from an arcade game.
It costs katex is not defined to play the game. Abbie plays the game katex is not defined times.
Calculate the profit or loss that Abby will make, stating clearly whether it is a profit or loss.
(5 marks)
Finding expenditure of katex is not defined pence or katex is not defined
katex is not defined(1)
Finding expected frequency of one of the amounts.
katex is not defined(1)
Finding one of the winnings.
katex is not defined – katex is not defined
katex is not defined pence or katex is not defined.
katex is not defined – katex is not defined
katex is not defined pence or katex is not defined.
katex is not defined – katex is not defined
katex is not defined or katex is not defined pence.
(1)
Finding sum of winnings.
katex is not definedkatex is not defined or katex is not defined pence
(1)
Stating “loss” and katex is not defined or katex is not defined pence.
katex is not defined(1)
Learning checklist
You have now learned how to:
- Understand probability distributions
- Use a probability model to predict the outcomes of future experiments
The next lessons are
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