How Do You Calculate the Expected Number of Occurrences?

Multiply the number of trials by the probability of the event occurring in each trial. If there are $n$ trials and the probability of success is $p$, the expected number of occurrences is $E=np$.

The Reasoning

An expected number describes the long-run average number of times an event should occur if the situation is repeated many times. It is not a guarantee of what will happen in one particular set of trials.

Use these steps:

1. Identify the total number of trials, $n$.
2. Identify the probability of the event in one trial, $p$.
3. Convert a percentage probability into a decimal if necessary.
4. Calculate (np).

For example, suppose $200$ students each have a probability of $0.15$ of choosing a particular activity. The expected number choosing it is

$$E=200(0.15)=30.$$

Therefore, approximately $30$ students are expected to choose the activity.

An expected value does not have to be a whole number. If $E=2.4$, this does not mean that $2.4$ occurrences are physically possible. It means that the average would approach $2.4$ occurrences per set of trials over many repetitions.

A common misconception is that the expected number must occur. It is only a theoretical long-run average, so the actual number may be higher or lower.

Exam Technique

This is SL 4.5: Trial and outcome, common to AI SL and HL, and may appear in either Paper 1 or Paper 2; a GDC is permitted. Show the substitution $E=np$, give the numerical result, and interpret it in context. Do not confuse an expected number with a probability: probabilities lie between $0$ and $1$, but an expected number of occurrences can be greater than $1$.