Expected frequency

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Expected Frequency

Here we will learn about expected frequency, including what it is and how to calculate it.

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 expected frequency?

Expected frequency is the number of times we would expect an event to occur over a given number of trials that take place during an experiment.

To find the expected frequency, we multiply the probability of that event taking place by the number of trials of the experiment.

\text{Expected frequency = probability of event }\times \text{ number of trials}

For example,

The probability of a spinner landing on a $‘2’$ is $0.3.$

Therefore if the spinner is spun $500$ times, the expected frequency of getting a $‘2’$ is

$0.3\times 500=150$ .

We would expect to get a $‘2’ \ 150$ times.

In reality, it is unlikely that the spinner will land on $‘2’$ exactly $150$ times as it is down to chance but this is a best estimate.

What is expected frequency?

Expected Value

You may need to use the expected frequencies to find an expected value.

For example,

A game at an amusement arcade has different cash prizes.

The probabilities for winning each cash prize are shown in the table.

It is $£1$ to play the game and Jonny decides to play the game $40$ times.

What is the expected value that Jonny will win in total?

Jonny expects to win $£0$ on $0.4 \times 40=16$ occasions.

He expects to win $50p$ on $0.4 \times 40=16$ occasions. $16 \times 50p=£8.$

He expects to win $£1$ on $0.15 \times 40=6$ occasions. $£1 \times 6=£6.$

He expects to win $£5$ on $0.05 \times 40=2$ occasions. $£5 \times 2=£10.$

In total he expects to win $£8 + £6 + £10 = £24.$

As he spent $£40$ playing the game, he will make a loss of $£40 - £24 = £16$ .

How to find expected frequency

In order to find the expected frequency of an event:

Determine the total number of trials of the experiment.
Determine the probability of the event.
Multiply the probability of the event by the number of trials of the experiment.

Explain how to find expected frequency

Probability distribution worksheet (includes expected frequency)

Get your free expected frequency worksheet of 20+ probability distribution questions and answers. Includes reasoning and applied questions.

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Probability distribution worksheet (includes expected frequency)

Get your free expected frequency worksheet of 20+ probability distribution questions and answers. Includes reasoning and applied questions.

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Related lessons on probability distribution

Expected frequency 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:

Expected frequency examples

Example 1: decimal probability

The probability that Sam will win a game of tennis is $0.7.$ If Sam plays $80$ games of tennis, how many can he expect to win?

Determine the total number of trials of the experiment.

The number of trials is $80.$

2Determine the probability of the event.

The probability of Sam winning is $0.7.$

3Multiply the probability of the event by the number of trials of the experiment.

Expected frequency $=0.7\times 80=56.$

Sam can expect to win $56$ games.

Example 2: fractional probability

The probability that a biased coin will show tails is $\frac{3}{8}.$ If the coin is flipped $400$ times, how many times can it be expected to show tails?

Determine the total number of trials of the experiment.

The number of trials is $400.$

Determine the probability of the event.

The probability of the coin showing tails is $\frac{3}{8}.$

Multiply the probability of the event by the number of trials of the experiment.

Expected frequency of tails $= \frac{3}{8}\times 400=150.$

Example 3: percentage probability

The probability that a biased $6$ sided die will land on a $5$ is $35\%.$ If the die is rolled $200$ times, estimate how many times it will not land on $5?$

Determine the total number of trials of the experiment.

The number of trials is $200.$

Determine the probability of the event.

The probability of the die not landing on $5$ is $100\%-35\% = 65\%.$

Multiply the probability of the event by the number of trials of the experiment.

Expected frequency of not $5 = 65\%$ of $200 = 130.$

Example 4: calculating expected frequency from another frequency

There are a number of coloured counters in a bag. This table shows the probability of picking each colour counter from the bag.

Lucy performs an experiment where she takes a counter, notes its colour and replaces it.

She repeats this experiment a number of times.

Lucy picks out a yellow counter $12$ times. Estimate how many times Lucy picks out a red counter.

Determine the total number of trials of the experiment.

We know that there are $12$ yellow counters and the probability of selecting a yellow counter is $0.2.$ Use the formula

$\text{Expected frequency = probability of event }\times \text{ number of trials}$ .

$12=0.2\times \text{number of trials}$

$\text{Number of trials} = \frac{12}{0.2}=60$

Determine the probability of the event.

The probability of selecting red is $0.15.$

Multiply the probability of the event by the number of trials of the experiment.

Expected number of red counters $=0.15\times 60=9.$

Common misconceptions

Dividing the number of trials by the probability

A common error is for students to divide the number of trials by the probability to find the excerpt frequency. It is important to think of the probability as a proportion of the number of trials.

If $P(red) = 0.3,$ it means that $30\%$ of the number of trials would be expected to be red, therefore if there were $200$ trials we would need to find $30\%$ of $200.$

Practice expected frequency questions

15

24

37.5

30

0.8 \times 30=24

333

40

100

120

\frac{3}{5} \times 200=120

5440

1440

8000

11111.11

36\%\,  of \,\ 4000 = \frac{36}{100} \times 4000 = 1440

1220

780

5128

3279

0.61 \times 2000=1220

45

600

400

500

\text{Expected frequency = probability of event }\times \text{ number of trials}

\begin{aligned} 150&=0.3 \times n\\\\ 150 \div 0.3 &=n\\\\ 500 &=n \end{aligned}

80

36

32

40

\begin{aligned} 12 &=0.15 \times n\\\\ 12 \div 0.15 &=n\\\\ 80 &=n \end{aligned}

The total number of trials was $80.$

Number of red counters $= 0.4 \times 80=32.$

Expected frequency GCSE questions

1. The probability that a school bus will be late is $0.2.$

Estimate the number of times the school bus will be late over a period of $80$ days.

(2 marks)

Show answer

0.2 \times 80

(1)

16

(1)

2. The probability that a biased coin lands on heads or tails is shown in this table.

Expected frequency GCSE question 2

The coin is flipped $600$ times. Estimate the number of times it will show tails.

(2 marks)

Show answer

0.46 \times 600

(1)

276

(1)

3. A bag contains a number of counters. This table shows the probability of picking each colour counter from the bag.

A counter is picked $400$ times.

Calculate an estimate for the number of times a yellow or blue counter is picked.

(3 marks)

Show answer

0.05+0.3=0.35

(1)

$0.35 \times 400$ or equivalent calculation.

(1)

140

(1)

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

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