Relative and expected frequencies

When outcomes are not equally likely

In the last lesson every probability came from counting equally likely outcomes: a fair dice, a fair spinner, a counter taken at random. Many real probabilities cannot be found that way.

• A drawing pin can land point up or point down, but the two are not equally likely, and its shape does not tell you the probability.
• A dice or spinner may be biased, so its faces are not equally likely.
• The probability that a seed grows, a bus is late or a customer buys a drink depends on the real world, not on symmetry.

For these you estimate the probability from data: the results of an experiment, or past records, or a survey.

Relative frequency

Each go at an experiment is called a trial. The relative frequency of an event is the fraction of the trials in which it happened:

$$\text{relative frequency}=\frac{\text{number of times the event happens}}{\text{total number of trials}}$$

For example, a spinner is spun 50 times and lands on 4 eleven times. The relative frequency of a 4 is $\frac{11}{50}=0.22$.

Relative frequency is an estimate of probability. Spin the same spinner another 50 times and you will probably get a slightly different count, so a relative frequency is never exactly the probability. But the more trials you do, the less the result is affected by luck, and the closer the relative frequency tends to get to the true probability. So:

• the more trials, the more reliable the estimate;
• a relative frequency from only a few trials can be a long way out.

Like any probability, a relative frequency is between 0 and 1 and can be written as a fraction, a decimal or a percentage. The bottom of the fraction is the total number of trials, not the number of times something else happened.

More trials, a steadier estimate

A coin was thrown 100 times and the relative frequency of heads was worked out after every 10 throws. Early on it jumps about (0.7 after 10 throws), but as the number of throws grows it settles close to 0.5, the probability for a fair coin. The value after the most trials is the best estimate.

Combining results

Sometimes several people carry out the same experiment, each doing a different number of trials.

• The single most reliable set of results is the one with the most trials.
• The best estimate of all uses every result together: add up all the times the event happened and divide by the total of all the trials.

Do not just find the mean of the separate relative frequencies. That gives a set of 10 trials the same weight as a set of 500, which is why it gives the wrong answer when the numbers of trials are different.

Fair, biased and random

These three words appear in many questions, and you must know exactly what they mean.

• Fair: every outcome is equally likely. A fair six-sided dice gives each number a probability of $\frac{1}{6}$.
• Biased: the outcomes are not equally likely. A biased dice might land on 6 more often than on 1, perhaps because it is weighted.
• Random: chosen in a way that gives each possible outcome, or each member of a group, the stated chance, with nobody choosing and no pattern. Picking names out of a hat without looking is random; picking the people nearest you is not.

Deciding whether something is biased. Compare the results with what a fair object would give. In 600 rolls of a fair dice you would expect about 100 of each number. Small differences, such as 94 or 107, are normal luck. A large difference over many trials, such as 160 sixes, is good evidence of bias. With only a few trials you cannot tell: getting no sixes in 10 rolls happens quite often with a fair dice.

A fair coin has no memory. After five heads in a row, the probability of heads on the next throw is still $\frac{1}{2}$. Each throw is a new trial, and the earlier results do not change it.

Expected frequency

If you know (or have estimated) the probability of an event, you can work out how many times it should happen in a number of trials:

$$\text{expected frequency}=\text{probability}\times \text{number of trials}$$

Why this works. A probability of 0.3 means the event happens in about 0.3 of the trials in the long run. In 200 trials that is $0.3\times 200=60$ times.

An expected frequency is what you would get on average. It is not a promise: in one set of 200 trials you might get 55 or 64. It does not have to be a whole number either; $0.3\times 25=7.5$ is a correct expected frequency.

The same idea estimates a number in a population. If a random sample shows that 15% of people do something, then about 15% of the whole population do it too. Use the relative frequency from the sample as the probability, then multiply by the size of the population.

Expected frequencies also predict money. A stall that charges for a game expects to take the cost of a go multiplied by the number of players, and to pay out the prize multiplied by the expected number of winners. The difference is the expected profit.