Relative frequency: worked examples and method

What it is

Relative frequency is the number of times an outcome happens divided by the total number of trials. It tells you what fraction of the experiment gave that outcome.

Work one out

A drawing pin is dropped 50 times and lands point up 18 times. The relative frequency of point up is 18/50 = 0.36. Always divide by all the trials.

An estimate

When outcomes are not equally likely you cannot count to find a probability. The relative frequency from an experiment is the best estimate you have.

More trials

The more trials you carry out, the more reliable the estimate becomes. A few trials are easily affected by luck, so they can give a result far from the true probability.

Is it fair

Compare the relative frequency with the fair probability. A fair dice shows a six one time in six, so in 300 rolls you expect 50 sixes. A big difference over many trials suggests the dice is probably biased.

Expect and miss

Expected number = probability x number of trials. All relative frequencies add up to 1, so a missing one is 1 minus the others. Expected is a prediction, not a promise.

Working out a relative frequency

A relative frequency compares one outcome with the whole experiment. Start by finding the total number of trials. If the results are in a frequency table, add up the whole frequency column. Then divide the frequency of the outcome you want by that total. A spinner that is spun 200 times and lands on green 68 times has a relative frequency of 68/200 = 0.34 for green. You can also write this as 34% or as 17/50. Check what form the question asks for.

The total must be the total number of trials, not the total of some of the outcomes. In a table with four colours, the relative frequency of green is green out of all four colours, not green out of green and red. A good check is to work out the relative frequency of every outcome. Together they must add up to 1. If they do not, the total was wrong or an outcome was missed.

Method: 1. Add up the frequencies to find the total number of trials. 2. Divide the frequency of the outcome by the total number of trials. 3. Give the answer in the form asked for, as a fraction, a decimal or a percentage. 4. Check that the relative frequencies of all the outcomes add up to 1.

Tip: Write the sum as a fraction first, such as 68 over 200, and only then turn it into a decimal. The fraction shows your method.

Using it to estimate a probability

When the outcomes are not equally likely, you cannot find the probability by counting. A relative frequency from an experiment is then the best estimate you have. If a biased dice is rolled 120 times and shows a six 30 times, the estimated probability of a six is 30/120 = 0.25. It is only an estimate. A second experiment with the same dice would give a slightly different answer.

The number of trials matters. After only 10 rolls a result is easily affected by luck, so the relative frequency can be far from the true probability. After 1000 rolls the luck mostly cancels out and the relative frequency settles close to the true value. In an exam, if you are asked how to make an estimate more reliable, the answer is to increase the number of trials. If you are asked which of two experiments gives the better estimate, choose the one with more trials.

Method: 1. Write the relative frequency of the outcome as the estimated probability. 2. Say that it is an estimate, because it comes from an experiment. 3. To improve the estimate, repeat the experiment with more trials.

Tip: The phrase to remember is more trials, better estimate. It answers most questions about reliability.

Is it fair? Comparing with what you expect

To test whether a dice, coin or spinner is fair, compare the result with what a fair one would give. A fair dice shows a six one time in six. In 300 rolls you would expect 300 ÷ 6 = 50 sixes. If the dice shows 72 sixes, that is 22 more than expected, and the relative frequency is 72/300 = 0.24 compared with 1/6 ≈ 0.17. The gap is large and the number of rolls is large, so the dice is probably biased.

Be careful with the wording. A small difference does not prove that something is biased, because chance always causes some difference. A big difference over many trials is good evidence of bias, and a big difference over only a few trials is not. Say that the dice is probably biased, or likely to be biased, never that it is definitely biased. Compare the relative frequency with the theoretical probability, and mention the number of trials in your reason.

Method: 1. Work out the theoretical probability for a fair version. 2. Work out the relative frequency from the experiment. 3. Compare the two, taking the number of trials into account. 4. Write a conclusion that says probably fair or probably biased, with a reason.

Predicting how many, and finding a missing value

Once you have a probability, or a relative frequency to use as one, you can predict how often an outcome will happen. Multiply the probability by the number of trials. If the probability that a bus is late is 0.15, then in 120 journeys you would expect 0.15 × 120 = 18 late buses. The word expect matters. It means the best prediction, not what will certainly happen. The actual number could easily be 16 or 20.

Tables of relative frequency often have one value missing. Because the relative frequencies of all the outcomes add to 1, the missing value is 1 minus the sum of the others. With values 0.3, 0.25 and 0.15 and one missing, the missing one is 1 - 0.3 - 0.25 - 0.15 = 0.3. You can then use it to predict. If the spinner is spun 80 times, the expected number for that outcome is 0.3 × 80 = 24.

Method: 1. If a value is missing, take the sum of the known relative frequencies away from 1. 2. Multiply the probability by the number of trials to find the expected number. 3. Write the answer as a whole number of times where the context needs one.

Tip: Show the multiplication in the working, such as 0.15 times 120. It earns the method mark even if the arithmetic slips.