Expected frequency Revision Notes for AQA GCSE Statistics

Expected frequency

What is expected frequency?

Expected frequency helps you predict how many times you expect an event to happen when you repeat an experiment a certain number of times. Think of it as making an educated guess based on the probability of an event occurring.

When you know the probability of something happening and you're going to repeat the experiment many times, expected frequency tells you roughly how many successes you should anticipate. However, it's important to understand that this is what you predict will happen - the actual results might be slightly different.

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The key distinction to remember is that expected frequency is a prediction based on probability theory, while actual frequency is what really happens when you perform the experiment. These two values are often close but rarely identical.

The key formula

The formula for calculating expected frequency is straightforward:

$\text{Expected frequency} = P(A) \times \text{number of trials}$

Where:

$P(A)$ is the probability of event A happening
Number of trials is how many times you repeat the experiment

This formula works because you're essentially saying "if this event happens with this probability each time, and I repeat it this many times, how many successes should I expect?"

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This formula only works when each trial is independent - meaning the outcome of one trial doesn't affect the others. This is true for dice rolls, coin flips, and spinner games, but not for situations like drawing cards without replacement.

Step-by-step worked example

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Worked Example: Rolling a Dice

Problem: A fair six-sided dice is rolled 100 times. Work out an estimate for the number of times you would expect to roll a number greater than 4.

Step 1: Identify the favourable outcomes Numbers greater than 4 on a dice are: 5 and 6 So there are 2 favourable outcomes out of 6 possible outcomes.

Step 2: Calculate the probability $P(\text{number} > 4) = \frac{2}{6} = \frac{1}{3}$

Step 3: Apply the expected frequency formula $\text{Expected frequency} = P(\text{number} > 4) \times \text{number of trials}$ $\text{Expected frequency} = \frac{1}{3} \times 100 = 33.33...$

Step 4: Round appropriately Since we can't roll a dice a fraction of a time, we round to 33.

Answer: You would expect to roll a number greater than 4 approximately 33 times.

Understanding expected frequency in practice

It's crucial to remember that expected frequency gives you a prediction, not a guarantee. If you actually rolled the dice 100 times, you might get 31 sixes and fives, or 35, or even 29. The expected frequency of 33 is your best estimate based on the probability.

The more trials you perform, the closer your actual results will typically be to the expected frequency. This is why expected frequency is particularly useful for planning and making predictions in situations involving many repetitions.

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Common Exam Tips

Always show your working clearly: Write out the probability calculation first, then multiply by the number of trials.
Round sensibly: You can't have fractional occurrences in real situations, so round to the nearest whole number unless told otherwise.
Don't overcomplicate fractions: You can often work with the fraction form rather than converting to decimals immediately.
Check your probability makes sense: Probabilities must be between 0 and 1, and your expected frequency should be less than or equal to your total number of trials.
Read the question carefully: Make sure you identify exactly what event you're calculating the expected frequency for.

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Key Points to Remember:

Expected frequency = Probability × Number of trials
It's a prediction of what should happen, not what will definitely happen
Always round your final answer appropriately for the context
The more trials you do, the closer actual results usually get to expected frequency
Show all your working clearly in exam questions - calculate probability first, then multiply by trials

Expected frequency (AQA GCSE Statistics): Revision Notes