C9.2; E9.2 - Relative and expected frequencies

C9.2; E9.2 - Relative and expected frequencies

A spinner lands on blue 14 times in 40 spins. Can this tell you how many blue results to expect in another 200 spins? C9.2 and E9.2 use experimental results to estimate probability, turn probabilities into expected counts and distinguish random variation from evidence of bias.

Relative frequency

When the probabilities are unknown, repeat the experiment under the same conditions and record the results. The proportion of trials in which an event occurs is its relative frequency.

Relative frequency

Relative frequency of an event=number of times the event occurstotal number of trials.\text{Relative frequency of an event}=\frac{\text{number of times the event occurs}}{\text{total number of trials}}.

This is an estimate of the event's probability, based on the observed trials. It is not automatically the exact probability.

For the opening spinner, 14 of 40 results were blue. The observed proportion is 14/4014/40. Repeating 40 spins could give a different count; an estimate describes the evidence so far.

Worked example 1: estimate from one result count

Question. A spinner lands on blue 14 times in 40 spins. Use these results to estimate the probability of blue.

Plan. Divide the observed blue count by all spins.

Estimated probability of blue=1440=720=0.35.\text{Estimated probability of blue}=\frac{14}{40}=\frac7{20}=0.35.

Answer. 0.350.35, or 720\frac7{20} or 35%35\%.

Check. The other 26 spins give relative frequency 26/40=0.6526/40=0.65, and 0.35+0.65=10.35+0.65=1. The estimate has the right probability scale.

The next event includes two score categories. Add their observed counts before dividing by the total number of trials.

Worked example 2: combine observed categories

Question. A die is rolled repeatedly, giving these results. Estimate the probability of a score greater than 4.

Score123456
Frequency91310121620

Plan. Count scores 5 and 6, then compare with all recorded rolls.

Number of rolls=9+13+10+12+16+20=80Number greater than 4=16+20=36Estimated probability=3680=920=0.45\begin{aligned} \text{Number of rolls}&=9+13+10+12+16+20=80\\ \text{Number greater than 4}&=16+20=36\\ \text{Estimated probability}&=\frac{36}{80}=\frac9{20}=0.45 \end{aligned}

Answer. 0.450.45.

Check. Scores 1 to 4 total 44, giving 44/80=0.5544/80=0.55. The estimated probabilities of the event and its complement add to 1.

Do not replace observed frequencies with an assumed fair model. A die with six faces need not have probability 1/61/6 for each face when the question asks you to use its experimental results.

Watch out

36/236/2 divides by the number of favourable category names. Relative frequency uses all 80 trials in the denominator, so the estimate is 36/8036/80.

Use all the trials, including the unsuccessful ones.

Expected frequency

A probability is a proportion. To estimate a count in a stated number of trials, multiply the proportion by that number. If the probability came from an experiment, the predicted count inherits that uncertainty.

Expected frequency

Expected frequency=probability×number of trials or objects.\text{Expected frequency}=\text{probability}\times\text{number of trials or objects}.

Use a probability appropriate to the trials or population being considered.

Not in the supplied formula list: learn it.

For a probability 0.350.35, the model expects 35 successes per 100 trials on average. It need not produce exactly 35 in a particular group of 100. An expected frequency can be non-integer: 8×0.35=2.88\times0.35=2.8 is a valid expectation, although an actual observed count is a whole number.

Worked example 3: a stated probability

Question. The probability that a randomly selected seed germinates is 0.180.18. Find the expected number that germinate from 200 seeds grown under these conditions.

Plan. Take 18 hundredths of the 200 seeds.

Expected number=0.18×200=36.\text{Expected number}=0.18\times200=36.

Answer. 36 seeds are expected to germinate.

Check. 36/200=0.1836/200=0.18, so the expected count gives the original proportion. It is less than 200, as required.

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