IGCSE Maths / Probability

Expected frequency and relative frequency

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

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Relative frequency uses experimental results to estimate a probability. You can use that estimate to judge whether an object may be biased and to predict how often an outcome may happen in a new set of trials.

How to find relative frequency

Relative frequency is the number of times an outcome happened divided by the total number of trials. It gives an estimate of the probability, because a different set of trials could give a different result. Write relative frequency as a fraction: frequency of the outcome ÷ total trials. In an exam, call it an estimate or estimated probability, not simply the probability.

How to check whether something may be biased

For a fair spinner with four equal sections, each colour would have probability 1/4 = 0.25. Compare the experimental relative frequency with this fair share. A large difference is evidence that the spinner may be biased. Blue appeared 57 times in 150 spins, so its relative frequency is 57/150 = 0.38. Since 0.38 is well above 0.25, the results are evidence that the spinner is biased. Yellow is also well below the fair share.

How to calculate an expected frequency

An expected frequency is a prediction for how often an outcome will happen in a new set of trials. Multiply its estimated probability by the new number of trials. Use the new number of trials, not the number already completed.

When the experiment is the only guide

Some probabilities cannot be worked out from theory. For a drawing pin, there is no fair share between point up and point down that tells you the chance of either outcome. The relative frequency from the experiment is the estimate to use. A pin landed point up 72 times in 200 drops, giving a relative frequency of 72/200 = 0.36. For 500 drops, the expected number landing point up is 0.36 × 500 = 180.

Apply the method to a dice

Use the experimental results to estimate the probability of a six, then scale that estimate to the number of rolls asked about. A dice landed on six 66 times in 300 rolls, so the estimated probability is 66/300 = 0.22. In 50 rolls, the expected number of sixes is 0.22 × 50 = 11.

More lessons in Probability

• The language of probability and the probability scale
• Listing outcomes systematically
• Theoretical probability and sample space
• Relative frequency and mutually exclusive events
• Multiplying independent probabilities
• At least one, and the complement
• Tree diagrams with replacement
• Tree diagrams without replacement
• Three stage tree diagrams
• Probability problem solving
• Conditional probability from a two way table
• Conditional probability from a Venn diagram
• Conditional probability without replacement
• Forming an equation from a probability