Probability: Expected and relative frequency

In a nutshell

Probabilities are usually theoretical value that indicate the chance of an event occurring, however probabilities can be estimated from relative frequency. The more times an experiment is run, the better an estimate of the true probabilities the relative frequencies are.

Expected frequency

Definition

The expected frequency of an outcome is the number of times the outcome is expected to occur. You can work it out using the following formula:

$\text{expected frequency} = \text{theoretical probability} \times \text{number of trials}$

Example 1

Ian is testing whether or not a coin is fair by flipping the coin $100$ times and recording the results. How many times should he expect the coin to land heads?

The probability of heads is $\dfrac 1 2$ . Use the formula to find the expected frequency.

Expected frequency = $\dfrac{1}{2} \times 100 = \underline{50}$ .

Relative frequency

Definition

The relative frequency of an outcome is the proportion of times the outcome has occurred out of all the outcomes. If an experiment is run for enough trials, it can be used as an estimate of the true probability of an outcome. You can work it out using the following formula:

$\text{relative frequency} = \dfrac{\text{frequency of outcome}}{\text{number of trials}}$

Example 2

Ian finds that from the $100$ flips, the coin lands heads $37$ times. What is the relative frequency of heads?

relative frequency = $\dfrac{37}{100} = \underline{0.37}$ .

Typical random experiments

Coin flip	Die roll	Matchsticks

Probability of: Heads $=\dfrac{1}{2}$ , tails $=\dfrac{1}{2}$ $\dfrac{$	Probability of: $P(1) =\dfrac{1}{6}$	Probability of: Short $=\dfrac{1}{3}$ , long $=\dfrac{2}{3}$