Expectation Formula. How to find the amount of times you expect an event to happen. Stats help.


The expectation is the predicted amount of times an event is going to happen. To work out the expectation you will need to know the probability of an event and the amount of trials (or times). All you need to do now is multiply these together:

Expectation = probability × number of trials

Let’s look at some examples of working out the expectation.

Example 1

David has a spinner. The probability of spinning a red on a spinner is 2/5. If he spins the spinner 30 times work out the expectation of spinning a red.

The probability of a red is 2/5 and the number of trials is 30, therefore:

Expectation = probability × number of trials

= 2/5 × 30 (type this in on your calculator or alternatively work out 2/5 of 30)

= 12

So the amount of times he is likely to spin a red is 12 times.

Example 2

The probability that is rains on any particular day is 0.15. Out of the next 200 days work out the amount of days that you expect it to rain.

The probability that it rains is 0.15 and the number of trials is 200, therefore:

Expectation = probability × number of trials

= 0.15 × 200

= 30

So the amount of days that it is likely to rain is 30 out of the 200 days.

Example 3

The probability that a person wins a prize on a raffle is 3/25. Work out the amount of prizes you will win if you make 50 entries on the raffle.

The probability that you win a prize is 3/25 and the amount of trials is 50. So all you need to do is multiply these values together:

Expectation = probability × number of trials

= 3/25 × 50

= 6

So the amount of prizes that you are likely to win is 6.

Example 4

A spinner is made up of 3 different colours: red, green and blue. The probability of spinning a red is 3/11 and the probability of spinning a green is 2/11. If the spinner is spun 132 times work out the expectation of a blue.

This question is a little trickier as you don’t have the probability of spinning a blue. You can find this by adding up the probabilities of spinning a red and green and subtracting this total from 1 (since probabilities always add up to 1):

3/11 + 2/11 = 5/11

1 – 5/11 = 6/11

So the probability of spinning a blue on the spinner is 6/11.

Also the number of trials is 132 so to find the expectation you need to multiply 6/11 by 132:

Expectation = probability × number of trials

= 6/11 × 132 (which is also the same as working out 6/11 of 132)

= 72

So the amount of times that you can expect to spin a blue out of the 132 is 72 times.

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