IGCSE Maths / Probability

Relative frequency and mutually exclusive events revision notes

Edexcel International GCSE Maths (4MA1), chapter 20: Probability, specification point 6.3d-f. Free lesson.

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Relative frequency uses experimental results to estimate probability and predict future outcomes. You can add probabilities to find the chance of one event or another only when the events cannot both happen.

How relative frequency estimates probability

The relative frequency of an outcome is how often it happened divided by the total number of trials. It estimates the probability, but it may not match the true probability exactly. Usually, more trials give a better estimate. For 100 spins, blue came up 31 times, so the estimated probability of blue is 31/100 = 0.31. A fair four-colour spinner would have probability 1/4 = 0.25 for each colour. Blue was above 0.25 and yellow was below it, which is evidence that the spinner may be biased. However, 100 spins is not many, so the results are not conclusive.

How to predict future results

Use the experimental estimate when predicting what happens in more trials. Multiply the estimated probability by the number of future trials. Here, the results suggest using 0.31 rather than the fair value of 0.25. The data is evidence that the spinner is not fair, so 0.31 is a better estimate for the prediction. The prediction is an expected number, not a guarantee of exactly how many times the outcome will happen.

When you can add probabilities

Mutually exclusive events cannot both happen on the same trial. For these events, add their probabilities to find the probability of one or the other. A single spin lands on only one colour, so it cannot be both red and blue. P(red or blue) = P(red) + P(blue) = 0.22 + 0.31 = 0.53. The answer is still a probability, so it must be no greater than 1. You cannot always add probabilities. If two events can happen together, adding them counts their overlap twice and makes the result too large. For example, it can both rain and be Monday.

Estimate the probability of green or yellow

Green and yellow cannot both happen on one spin, so their relative frequencies can be added. Green has relative frequency 0.27 and yellow has relative frequency 0.20. P(green or yellow) = 0.27 + 0.20 = 0.47.

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