23.2 Relative frequency and expected outcomes — Learn

Prerequisites: fractions, percentages and multiplication.

Objectives: calculate relative frequency; use experimental results to estimate probability; calculate expected counts.

Relative frequency is the number of times an event happens divided by the total number of trials. A trial is one repeated attempt. Relative frequency estimates probability from data; it may change when more trials are added.

Example 1. A coin lands on heads 54 times in 100 tosses. Find the relative frequency of heads.

This result estimates the chance of heads. A fair coin has theoretical probability $\frac12$, but a sample need not give exactly 50 heads.

Expected frequency is the number of trials multiplied by the probability. It is a prediction, not a promise about the observed result.

Example 2. A fair six-sided die is rolled 240 times. Estimate the number of sixes.

The probability of a six is $\frac16$.

The expected number is 40 sixes.

Watch out. An expected frequency can be fractional, and an observed count can differ. For 10 trials with probability $\frac14$, the expectation is 2.5, not a guaranteed count.

Progress check

Suppose the spinner lands on red 32 times in 50 spins. Work out the relative frequency of red. Use it to estimate the number of reds in 200 spins. Does that agree with your first prediction of about half?

You should now be able to

• calculate relative frequency from results;
• explain that relative frequency estimates probability and can change with more trials;
• calculate expected outcomes from probability and trial count.