E8.2 Relative and expected frequencies

Learning objectives

E8.2.1Understand relative frequency as an estimate of probability.• Understand relative frequency as an estimate of probability.E8.2.2Calculate expected frequencies• Calculate expected frequencies. e.g. use results of experiments with a spinner to estimate the probability of a given outcome. e.g. use probability to estimate an expected value from a population. Includes understanding what is meant by fair, bias and random.

Relative frequency is the proportion of trials in which an event occurs. It gives an experimental estimate of the event's probability.

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

A spinner is used 80 times and lands on 5 on 13 occasions. The relative frequency of landing on 5 is $\tfrac{13}{80}=0.1625$ , so $0.1625$ is an estimate of the probability of landing on 5.

A different set of trials can give a different relative frequency. A larger, well-run sample usually gives a more stable estimate because one unusual result has less influence, but it does not guarantee the exact theoretical probability.

To combine experiments, add the event counts and add the trial counts, then divide the two totals. For example, results of 8 successes from 30 trials and 17 from 70 trials combine to $\tfrac{8+17}{30+70}=0.25$ . Do not take an unweighted average when the trial totals differ.

Relative frequency is evidence from observed results, not a certainty about the next result. 'Random' means an individual result is unpredictable; it does not mean that every outcome must have equal probability.

Expected frequency is the number of occurrences predicted from a probability over a stated number of trials or members of a population.

\text{expected frequency}=\text{probability of the event}\times\text{number of trials or population size}

A bag contains 7 red, 5 green and 2 pink counters. A counter is selected at random, replaced and the experiment is repeated 140 times. Since $P(\text{green})=\tfrac5{14}$ , the expected frequency of green is $\tfrac5{14}\times140=50$ . Replacement keeps the probability the same for each trial.

If the probability is unknown, estimate it with relative frequency first. In a representative sample, 36 of 240 people have a feature, so the estimated probability is $\tfrac{36}{240}=0.15$ . For a population of 1600, the expected number is $0.15\times1600=240$ .

Term	Meaning
fair	the relevant outcomes have equal probabilities
biased	the mechanism favours some relevant outcomes, so their probabilities are not equal
random	an individual result cannot be predicted with certainty; probabilities need not be equal

An expected frequency is a long-run prediction, not a guarantee of the actual count. Keep the probability unrounded until the final multiplication. A calculation may give a non-integer expectation even though an observed count must be a whole number; round only when the context or question requires it.

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—E8.3 Probability of combined eventsRead

E8.2 Relative and expected frequencies

Learning objectives

Use relative frequency to estimate probability

Calculate expected frequency and interpret fair, biased and random

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