Expected Probabilities

Video coming soon3 worked examples

Theory

Probability doesn't just tell us the chance of a single event happening; it allows us to predict the future on a larger scale. Expected probabilities (also known as expected values or expected success) are used to predict how many times we expect a specific outcome to happen over a large number of trials.

1. The Expected Success Formula

To calculate an expected value, you simply multiply the probability of the outcome by the total number of trials (the sample size).

\text{Expected Success} = \text{Probability} \times \text{Sample Size}

Crucial Note:

Expected probability does not guarantee an exact, real-world result. Instead, it provides businesses and project managers with a strong mathematical indication of what is most likely to happen, allowing them to adjust their finances and schedules accordingly.

2. Expected Probabilities and the 'NOT' Rule

When calculating expected costs, profits, or successful days, you will often be given the probability of a failure (e.g., a bus being late, or a machine breaking).

Before you can calculate the expected number of successes, you must first calculate the probability that the failure does not happen using the rule: $P(\text{Not A}) = 1 - P(A)$ .

Worked examples

Example 1

Example 1: Basic Expected Success

A smartphone manufacturer tests a new batch of screens. The probability of a screen having a defective pixel is 0.015. If the factory produces 8,400 screens in a single day, how many screens are expected to be defective?

$\text{Expected Success} = \text{Probability} \times \text{Sample Size}$

$\text{Expected Defective Screens} = 0.015 \times 8400 = 126$ screens.

Example 2

Example 2: Using the 'NOT' Rule with Time

An outdoor cinema operates every single day throughout July and August (62 days in total). Based on meteorological data, the probability of heavy rain forcing the cinema to close on any given day is 0.15.

Calculate the expected number of days the outdoor cinema will be able to remain open.

First, find the probability that it does not rain (meaning the cinema is open):

$P(\text{Open}) = 1 - 0.15 = 0.85$

Next, calculate the expected number of open days:

$\text{Expected Days Open} = 0.85 \times 62 = 52.7$ days.

(The cinema can expect to be open for approximately 53 days).

Example 3

Example 3: Financial Forecasting

A car dealership sells 420 cars in a year. The probability that a customer chooses to add the optional "premium winter tyre package" to their purchase is 0.25. The profit made on each winter tyre package is £150.

Calculate the total expected profit the dealership will make from the winter tyre packages this year.

First, calculate how many customers are expected to buy the package:

$\text{Expected Customers} = 0.25 \times 420 = 105$ customers.

Next, calculate the expected profit:

$\text{Expected Profit} = 105 \times \text{\pounds}150 = \text{\pounds}15,750$ .

Ready to practise?

7 Expected Probabilities questions with answers and video solutions