0580 · 8.2Relative and expected frequencies

Relative frequency from results

0:00A spinner lands on red twenty-eight times in eighty spins. That result can help us predict future outcomes, even if we cannot tell whether the spinner is fair. But a prediction is not a promise about what happens next. In this video, we will use experimental results to estimate probability, then turn probabilities into expected numbers. An outcome is a possible result, such as landing on red. An event is the outcome, or group of outcomes, that interests us.

0:25Probability describes how likely an event is, using values from zero to one. We can write those values as fractions, decimals or percentages, depending on what makes the calculation clearest. A trial is one attempt at an experiment, such as one spin. The frequency of an event is the number of times it occurs. Relative frequency compares that count with the total number of trials. It equals the frequency of the event divided by the total number of trials.

0:50The division turns a count into a proportion. Twenty red results in forty spins mean something different from twenty in two hundred. In forty spins, red occurred in half the trials. In two hundred spins, the same count represents only one tenth, so the total number of attempts matters. Relative frequency is an estimate of probability based on experimental results. We use the proportion observed so far as a guide to what might happen under the same conditions.

1:14It describes the evidence from the experiment, not a guaranteed future pattern. Another set of trials may produce a different proportion. Suppose a spinner produces twenty-eight red results, twenty blue results and thirty-two yellow results. The question asks us to estimate the probability of landing on red. Adding twenty-eight, twenty and thirty-two gives eighty trials altogether. We need the red frequency divided by this total, because we want the proportion of spins that landed on red.

1:41Write the estimate as 28/80. Dividing the numerator and denominator by four gives 7/20. Converting this fraction gives zero point three five. Call the event of landing on red R. Our estimate of P of R is therefore zero point three five, based on the experimental results. This means red occurred in thirty-five per cent of the recorded spins. It does not mean the next hundred spins must contain exactly thirty-five red results.

2:07The estimate could change when we collect more data. Also, the next spin is not determined by the order of earlier outcomes.

Fairness, randomness and reliability

2:14A fair coin, dice or spinner gives every outcome the same chance. On an ideal spinner with four equal sectors, each sector is equally likely. Each colour here therefore has probability 1/4. A biased device is not fair, because some outcomes are more likely than others, perhaps because it is weighted or uneven. Random means that an individual outcome is governed by chance rather than being fixed in advance. It does not mean every possible result is equally likely.

2:41A biased spinner can still produce random outcomes, even though one colour has a greater probability. Fairness describes the probabilities, while randomness describes how individual results arise. Experimental results alone need careful interpretation when judging fairness. A fair spinner will not necessarily produce equal counts in a short experiment. Differences can happen simply through chance. A large, persistent difference from the equal proportions provides stronger evidence of bias, but a small difference does not prove it.

3:10A larger number of trials generally gives a better estimate of the probability, provided the conditions stay the same. In a small experiment, a few unusual outcomes can strongly affect the proportion. In a large experiment, each additional outcome has less influence. However, the estimate does not have to move closer to the true probability after every trial.

Calculating expected frequency

3:30Expected frequency is how many times we expect an event to occur in a given number of trials. To calculate it, multiply the probability by the number of trials. This reverses the relative frequency calculation. Dividing a count gives a proportion, while multiplying a proportion gives a predicted count. Think of probability as the share of trials we expect to produce the event. If the probability is one tenth, we expect about a tenth of all trials to produce the event.

3:57Multiplication finds that share of the total. The resulting number is an expectation, not a compulsory total. The spinner from our first example is spun a further two hundred and forty times. Use 7/20 as the estimated probability of red. The expected red frequency is 7/20 × 240. First, 240 ÷ 20 gives twelve. Multiplying seven by twelve gives eighty-four expected red results. We have used earlier results to predict a count in a new experiment.

4:26The prediction assumes that the spinner and the way it is used remain unchanged. We expect eighty-four red results, but the actual count may differ. Also, two hundred and forty is the new trial total, not the earlier eighty. For our second example, find the expected number of blue results when the fair four-colour spinner is spun three hundred times. Call the event of landing on blue B. Since exactly one sector is blue, P of B is 1/4.

4:50The expected blue frequency is 1/4 × 300. Dividing three hundred by four gives seventy-five expected blue results. This probability comes from the stated fair design, rather than an experimental estimate. Fairness still does not guarantee seventy-five blue results in this particular experiment. If the question gives experimental results instead, use that data when asked to estimate probability. Do not assume equal chances merely because a spinner has several colours.

Population estimates and exam tips

5:18Expected frequencies can also predict numbers within a population. A population is the whole group we are interested in, such as every student at a school. A sample is a smaller group used to gather information. Applying its proportion to the whole population assumes that the sample reasonably represents that population. Suppose a random sample of two hundred students contains fifty-four who walk to school. The school has one thousand five hundred students altogether.

5:43Estimate how many students in the whole school walk there. Begin with 54/200, because this is the proportion walking in the sample. Simplifying 54/200 gives 27/100, which is zero point two seven. Multiply 27/100 by one thousand five hundred. Dividing one thousand five hundred by one hundred gives fifteen. Then 27 × 15 gives four hundred and five. We estimate that four hundred and five students walk to school. The population prediction depends on the sample being representative.

6:13Asking only students whose homes are beside the school could overestimate the proportion who walk. A larger sample from that same restricted group would not automatically fix the problem. Random selection helps avoid deliberately choosing students with particular characteristics. A common mistake is dividing the total number of trials by the event frequency. That gives trials per occurrence, rather than the required proportion, and may produce a value greater than one.

6:38Another mistake is using the old trial total when predicting future results. Identify the event count, the original trial total and the new one separately. An expected frequency does not always have to be a whole number. For example, a probability of zero point three across twenty-five trials gives an expected frequency of seven point five. The actual count must be a whole number, but the expectation can lie between two integers.

7:02Only round it to a whole number when the question or context requires that. These methods must work without a calculator on Paper two, and they also appear on calculator Paper four. Simplifying fractions before multiplying often makes the arithmetic easier. Keep exact values throughout the working, rather than rounding an estimated probability too early. For answers that are not exact, use three significant figures unless instructed otherwise. An examination may ask you to calculate a relative frequency and then calculate an expected frequency.

7:30Show the event frequency divided by the original trial total first. Multiply that probability by the specified new total to obtain the prediction. If asked to explain greater reliability, state that more trials generally reduce the effect of chance variation on the proportion. The central connection is between counts and proportions. Relative frequency divides an observed count by the total to estimate probability. Expected frequency multiplies a probability by a trial total or population size to predict a count.

7:59Keep fairness separate from randomness, and remember that an expectation describes what we anticipate, not what must happen.