C8.2; E8.2 - Relative and expected frequency

C8.2; E8.2 - Relative and expected frequency

An experiment can suggest a probability without revealing its exact value. For Core and Extended, C8.2/E8.2 uses relative frequency to estimate chance and expected frequency to predict counts. You will also distinguish fair, biased and random processes, and explain why a prediction is not a guarantee.

Experimental relative frequency

A coin gives seven heads in ten tosses. Has it proved that the coin is biased? A short run can be uneven by chance, even for a fair coin. We can describe what happened exactly while remaining cautious about the probability behind it.

Relative frequency

The relative frequency of an event is

relative frequency=number of times the event occurredtotal number of trials.\text{relative frequency}=\frac{\text{number of times the event occurred}}{\text{total number of trials}}.

It is an experimental estimate of the event's probability, rather than automatically its exact theoretical probability.

For the coin, the relative frequency of heads is 7/10=0.77/10=0.7. This is a fact about the ten recorded tosses. It does not prove that the chance of heads on the next toss is 0.70.7.

To use results to estimate future chance, the trials should represent the process you want to predict. Keep the conditions stable: a spinner whose pointer is altered halfway through is no longer one unchanged experiment. A larger set of representative trials generally gives a more dependable estimate by reducing the influence of a few unusual results. It need not be closer to the true probability after every extra trial.

Worked example 1: estimating from a trial count

Question. A spinner is spun 8080 times and stops on blue 2828 times. Estimate the probability of blue.

Plan. Divide the observed blue count by the total trials.

estimated probability=2880blue results over all spins=720=0.35simplify the observed proportion\begin{aligned} \text{estimated probability}&=\frac{28}{80} &&\text{blue results over all spins}\\ &=\frac7{20}=0.35 &&\text{simplify the observed proportion} \end{aligned}

Answer. An estimate is 0.350.35, or 35%35\%.

Check. 35%35\% of 8080 is 2828, recovering the observed number of blue results.

Keep the same spinner and change only the amount of evidence. Combine counts before finding the overall relative frequency.

Worked example 2: extending the experiment

Question. The first 8080 spins gave 2828 blue results. A further 120120 spins under the same conditions give 3838 blue results. Use all the results to estimate the probability of blue.

Plan. Add the two blue counts and the two trial counts, then form one fraction.

blue total=28+38=66all favourable resultstrial total=80+120=200all trialsestimated probability=66200=0.33overall relative frequency\begin{aligned} \text{blue total}&=28+38=66 &&\text{all favourable results}\\ \text{trial total}&=80+120=200 &&\text{all trials}\\ \text{estimated probability}&=\frac{66}{200}=0.33 &&\text{overall relative frequency} \end{aligned}

Answer. The combined estimate is 0.330.33.

Check. The non-blue count is 52+82=13452+82=134. The blue and non-blue relative frequencies add to 66/200+134/200=166/200+134/200=1.

The changed estimate is not a contradiction: the first estimate used less evidence. Do not simply average two relative frequencies when their trial totals differ. A result based on 120120 trials should carry more weight than one based on 8080.

Watch out

“The relative frequency is 28/5228/52” compares blue results with non-blue results. The denominator must include every trial: 28/(28+52)=28/8028/(28+52)=28/80.

Build one overall fraction from the two new runs below.

Expected numbers of outcomes

If the probability of red is 0.40.4, an estimate for the number of reds in 100100 similar trials is 4040. The multiplication scales a proportion to a count. It does not schedule which trials will be red or promise exactly forty.

Expected frequency

The expected frequency of an event in a stated number of trials is

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

Use an estimated probability when only experimental evidence is available. The prediction then also depends on that estimate being suitable for the future trials.

Why multiply? A probability of 0.40.4 represents an expected proportion of four tenths. Taking four tenths of the number of trials gives the corresponding expected count. The formula works with a known theoretical probability or an experimental estimate, but an experimental estimate does not become exact through multiplication. This relation is not supplied in the List of formulas.

Worked example 3: predicting a count

Question. The probability that a seed from a particular batch germinates is estimated as 0.80.8. Assuming that estimate applies, how many of 150150 similar seeds are expected to germinate?

Plan. Take eight tenths of the number of seeds.

expected frequency=150×0.8trials times estimated probability=120eight tenths of one hundred and fifty\begin{aligned} \text{expected frequency}&=150\times0.8 &&\text{trials times estimated probability}\\ &=120 &&\text{eight tenths of one hundred and fifty} \end{aligned}

Answer. 120120 seeds are expected to germinate.

Check. The expected non-germinating count is 150×0.2=30150\times0.2=30; 120+30=150120+30=150.

Keep the probability and change only the number of seeds. The predicted count scales in the same proportion.

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