## What you need to know

The relative frequency (or experimental probability) of something happening is the probability of that outcome based on data that’s been collected in an experiment. We calculate relative frequency using the following formula:

$\text{relative frequency }=\dfrac{\text{number of times an outcome happened}}{\text{total number of all outcomes}}$

For example, if you were to flip a coin 100 times and record the results, then you could say

$\text{relative frequency of landing on heads }=\dfrac{\text{number of heads}}{100}$

Now, obviously we know that even before flipping a normal coin, the chance of heads is going to be 50%. This is called the theoretical probability, and we can use it in this example because coins are easy to understand. However, theoretical probability isn’t always so obvious, so we have to conduct an experiment and find the relative frequency instead.

Example: Prim asks some people in her town about their dietary habits and records the results in the table below.

a) Work out the relative frequency of someone in Prim’s town being vegetarian.

b) There are 20,000 people in Prim’s town. Using your answer to part a), find an estimate for the number of people in this town who are vegetarian.

a) Firstly, we need the total number of all outcomes:

$\text{total }=44+23+43=110$

Now, using the formula shown above, we can calculate the relative frequency for vegetarians to be

$\text{relative frequency }=\dfrac{44}{110}=\dfrac{2}{5}$

b) Our answer to part a) suggests that if we picked a random person in Prim’s town, there would be a 2/5 or 0.4 chance that they are vegetarian. Another way of thinking about this is: we expect that if we ask a bunch of people, then 2/5 of them will be vegetarian.

To answer this question, we need to find 2/5 of 20,000 – that’s how many people we expect to be vegetarian. Doing this, we get

$\dfrac{2}{5}\times 20,000=8,000\text{ vegetarians}$

What we actually found here is called the expected frequency – you don’t need to worry about remembering this term, but the idea is important. 8,000 is probably not the *exact* number of vegetarians in the town, but it is the amount that we expect there to be based on the information that Prim collected. In other words, it’s our best estimate with what we’ve got.

Bias

One of the key uses of relative frequency is in testing for bias. We say that an experiment is biased when the probability of a particular outcome is unfairly bigger or smaller than it should be.

For example, a biased coin might be cleverly designed so that it’s lands on tails more than 50% of the time. If we suspect a coin of being biased, then we test it by flipping it a load of times and recording the results. Then, if the relative frequency of tails is noticeably more than 50%, we might suspect the coin of being biased.

Example: Amber has a spinner with the numbers 1, 2, 3 and 4 on it. She was told that the spinner is fair, i.e. there is an equal chance of landing on any number when you spin it, but she claims that the spinner is biased.

Amber spins the spinner 150 times and it lands on the number one 54 times. By calculating the relative frequency of 1, comment on Amber’s claim.

Firstly, the relative frequency:

$\text{relative frequency of landing on 1 }=\dfrac{54}{150}=0.36$

So, the relative frequency is 0.36, or 36%. But we know there are 4 equally likely outcomes on the spinner, meaning the theoretical probability of landing on 1 is

$\dfrac{1}{4}=25\%$

25% is significantly smaller than 36%, so it seems that Amber has good to reason to make the claim she does. The results suggests the spinner is biased.

NOTE: in reality we don’t know this for sure. It’s completely possible that a fair spinner could land on the number one 54 times, but it is unlikely, which leads us to believe the spinner is biased.

We can always do a better experiment and make our conclusion more confident by

increasing the number of trials!

In this case, that would mean spinning the spinner 200, 300, or even 1,000 times. The more, the merrier! If you spin it 1,000 times and still end up getting about 10% more number 1s than you should, then you can be even more confident the spinner is biased.

## Example Questions

a) 12 students used public transport to get to school, so to 3dp we get

$\text{relative frequency of travelling on public transport }=\dfrac{12}{62}=0.194$

b) 16 students walked to school, so to 3dp we get

$\text{relative frequency of walking }=\dfrac{16}{62}=0.258$

c) There were 62 students asked in total and 9 of them cycled, so the number who didn’t cycle is

$62-9=53$

Therefore, we get, to 3dp

$\text{relative frequency of not cycling }=\dfrac{53}{62}=0.855$

In Bev’s experiment, the relative frequency of rolling a 6 is

$\dfrac{63}{400}=0.1575$

Now, the theoretical probability of rolling a 6 is

$\dfrac{1}{6}=0.166666...$

The two results are quite close, in fact the relative frequency is slightly lower but nothing to be really noticeable.

So, we can conclude that Bev’s statement is false, the die does not appear to be biased.