Apply ideas of randomness, fairness and equally likely events to calculate…

The idea

This skill connects three ideas. Randomness means each trial is unpredictable, yet over many trials a pattern emerges. Fairness means no outcome is favoured, so a fair six-sided die gives each face probability 1/6. Equally likely events have the same probability, which lets you calculate expected outcomes by multiplying the probability of an outcome by the number of trials. For example, if a fair coin is tossed 200 times, the expected number of heads is 200 × 1/2 = 100. For multiple future experiments, apply the same expected value to each experiment and combine the totals, for example 3 experiments of 50 spins each give 150 spins in total. Expected outcomes describe long-run averages, not guaranteed results in any single experiment.

What earns the marks

• Identifies whether outcomes are equally likely and states the probability of each outcome for a fair object.
• Calculates an expected frequency by multiplying the probability of an outcome by the number of trials.
• Applies the same expected-value calculation across multiple future experiments and combines the totals correctly.
• Interprets an expected outcome as a long-run average rather than a certain result.
• Explains how randomness and fairness justify using equal probabilities in the calculation.
• Recognises that observed results may differ from expected results, especially in small numbers of trials.

Common mistakes and how to put them right

• Assuming an expected outcome must occur exactly.

  Put it right: Describe it as the average result over many trials, with individual experiments varying around it.

• Using unequal probabilities for a fair object, for example giving a fair die faces different probabilities.

  Put it right: Assign probability 1/6 to each face.

• Multiplying the number of trials by the number of outcomes instead of by the probability.

  Put it right: Use expected frequency = probability × number of trials.

• Forgetting to combine the trials when several future experiments are described.

  Put it right: Find the total number of trials first, or to calculate each experiment separately and add the expected outcomes.

Exam tips

• Write down the probability of the outcome you are using before multiplying, so your expected-value calculation is clear.
• When several experiments are described, state the total number of trials or show each expected outcome separately and then add them.
• Use the words expected or average in your final answer to show you understand it is a long-run prediction, not a guaranteed result.