Expected Outcomes

This lesson covers how to calculate the expected number of times an event will occur, given a probability and a number of trials. This is a key topic in the AQA GCSE Mathematics specification and frequently appears in exam questions combining probability with number work.

What Are Expected Outcomes?

If you know the probability of an event and the number of times an experiment is repeated, you can calculate the expected frequency (the number of times you expect the event to happen).

Expected frequency = probability x number of trials

This is sometimes called the expected number of outcomes or expected value.

Worked Example 1

A fair coin is flipped 200 times. How many times would you expect to get heads?

P(heads) = 1/2 = 0.5
Number of trials = 200

Expected frequency = 0.5 x 200 = 100

You would expect to get heads 100 times.

Exam Tip: The expected frequency is a theoretical prediction. In practice, you are unlikely to get exactly 100 heads in 200 flips — but it is the best estimate based on the given probability.

Calculating Expected Frequency from a Probability Table

Worked Example 2

A biased die has the following probability distribution:

Score	1	2	3	4	5	6
Probability	0.1	0.1	0.15	0.2	0.2	0.25

The die is rolled 400 times. How many times would you expect to score a 6?

Expected frequency of 6 = 0.25 x 400 = 100

How many times would you expect to score 3 or less?

P(3 or less) = P(1) + P(2) + P(3) = 0.1 + 0.1 + 0.15 = 0.35

Expected frequency of 3 or less = 0.35 x 400 = 140

Comparing Expected and Actual Results

Exam questions often ask you to compare the expected results with the actual (observed) results and comment on the difference.

Worked Example 3

A spinner has four sections: Red, Blue, Green, Yellow. Each section is the same size. The spinner is spun 80 times.

Colour	Expected Frequency	Actual Frequency
Red	20	23
Blue	20	17
Green	20	22
Yellow	20	18

Comment on the results.

The actual results are close to the expected results. The differences are small and can be explained by natural variation (randomness). This suggests the spinner is fair.

Exam Tip: When comparing expected and actual results, comment on whether the differences are small or large. Small differences suggest fairness; large differences (especially over many trials) suggest bias.

Working Backwards from Expected Frequency

Sometimes you are given the expected frequency and need to work backwards to find the probability or the number of trials.

Worked Example 4 — Finding the Probability

A bag of sweets contains red and yellow sweets. A sweet is picked at random 150 times (with replacement). A red sweet is expected to be picked 60 times.

What is the probability of picking a red sweet?

P(red) = expected frequency / number of trials = 60/150 = 2/5

Worked Example 5 — Finding the Number of Trials

The probability that a biased coin lands on tails is 0.6. The coin is flipped a certain number of times. The expected number of tails is 90.

How many times was the coin flipped?

Number of trials = expected frequency / probability = 90 / 0.6 = 150

Exam Tip: The formula expected frequency = probability x trials can be rearranged. Make sure you are comfortable solving for any one of the three quantities when given the other two.