Probability

Experimental Probability: Relative Frequency and Expected Outcomes

Year 7 · Year 8 · Year 9

✓By the end of this lesson students will be able to define experimental probability and relative frequency.
✓By the end of this lesson students will be able to calculate the relative frequency of an event from a set of experimental data.
✓By the end of this lesson students will be able to use relative frequency to estimate the probability of an event.
✓By the end of this lesson students will be able to calculate the expected number of outcomes for an event based on experimental probability.
✓By the end of this lesson students will be able to understand that increasing the number of trials generally leads to a more reliable estimate of probability.

Key concepts

Experimental Probability

Experimental probability is the probability of an event occurring based on the results of an experiment or a series of trials. It is determined by carrying out an experiment and observing the outcomes, rather than by theoretical calculations.

Relative Frequency

Relative frequency is the number of times an event occurs in an experiment, divided by the total number of trials conducted. It is used as an estimate for the probability of an event, especially when theoretical probability is difficult or impossible to calculate.

Relative Frequency = (Number of times the event occurs) / (Total number of trials)

Expected Outcomes

The expected number of outcomes for an event is the predicted number of times an event will occur if an experiment is repeated a certain number of times. It is calculated by multiplying the probability of the event by the total number of trials.

Expected Outcomes = Probability × Total number of trials

Key facts to remember

1Experimental probability is based on actual results from an experiment.
2Relative frequency is the ratio of the number of times an event occurs to the total number of trials.
3Formula for relative frequency: (Number of successful trials) / (Total number of trials).
4Relative frequency provides an estimate of the true probability of an event.
5The more trials conducted in an experiment, the more reliable the estimate of the probability will be.
6Expected outcomes are calculated by multiplying the probability of an event by the total number of trials.
7Relative frequency, like all probabilities, must be between 0 and 1 (inclusive).
8The sum of the relative frequencies for all possible outcomes in an experiment should be 1.

Worked examples

Example 1

A biased coin is tossed 50 times. It lands on heads 32 times. Calculate the relative frequency of landing on heads.

IIdentify the number of times the event (landing on heads) occurred: 32.

IIIdentify the total number of trials: 50.

IIIApply the relative frequency formula: Relative Frequency = (Number of heads) / (Total tosses).

IVSubstitute the values: Relative Frequency = 32 / 50.

VSimplify the fraction or convert to a decimal: Relative Frequency = 16 / 25 or 0.64.

Answer

0.64

Relative frequency can be expressed as a fraction, decimal, or percentage.

Example 2

A spinner has three colours: red, blue, and green. It is spun 200 times. The results are: Red: 85 times, Blue: 60 times, Green: 55 times. Estimate the probability of the spinner landing on blue.

IIdentify the number of times the event (landing on blue) occurred: 60.

IIIdentify the total number of trials: 200.

IIICalculate the relative frequency for blue: Relative Frequency (Blue) = (Number of blue outcomes) / (Total spins).

IVSubstitute the values: Relative Frequency (Blue) = 60 / 200.

VSimplify the fraction: Relative Frequency (Blue) = 3 / 10 or 0.3.

VIState the estimated probability: The estimated probability of landing on blue is 0.3.

Answer

0.3

The sum of the relative frequencies for all possible outcomes should be 1.

Example 3

Based on the experiment in Example 2, if the spinner is spun another 500 times, how many times would you expect it to land on red?

IFirst, calculate the relative frequency of landing on red from the initial experiment: Relative Frequency (Red) = (Number of red outcomes) / (Total spins) = 85 / 200 = 0.425.

IIThis relative frequency is our estimated probability for red.

IIIIdentify the new total number of trials: 500.

IVApply the expected outcomes formula: Expected Outcomes = Estimated Probability × Total number of trials.

VSubstitute the values: Expected Outcomes = 0.425 × 500.

VICalculate the result: Expected Outcomes = 212.5.

Answer

212.5 times

Expected outcomes do not have to be whole numbers, as they are theoretical predictions based on probability.

Common mistakes

✗Confusing experimental probability (based on trials) with theoretical probability (based on equally likely outcomes).
✗Incorrectly setting up the fraction for relative frequency (e.g., swapping numerator and denominator).
✗Assuming that experimental results will exactly match theoretical probabilities, especially with a small number of trials.
✗Rounding relative frequencies too early in calculations, leading to inaccuracies in expected outcomes.
✗Not understanding that expected outcomes can be non-integer values.

Exam tips

★Always show your working clearly, especially when calculating relative frequency and expected outcomes.
★Read the question carefully to determine if you need to calculate relative frequency, use it as an estimate, or find expected outcomes.
★Ensure your probability values (relative frequencies) are always between 0 and 1. If not, recheck your calculations.
★When asked to estimate probability, use the relative frequency from the given data.
★Remember that 'expected outcomes' are predictions and may not be exactly what happens in reality.

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum