Probability

Single and combined events, the AND/OR rules, tree diagrams and expected outcomes.

7 min read

Probability is measured from $0$ to $1$ . Most ESAT questions combine events, so the key skills are the AND/OR rules and laying out outcomes with a tree or table.

For equally likely outcomes, $P (event) = \frac{favourable outcomes}{total outcomes}$ . The probabilities of all outcomes sum to $1$ , so $P (not A) = 1 - P (A)$ .

Exam tip

When "at least one" appears, it is almost always faster to compute $1 - P (none)$ than to add up every case.

AND (independent events): multiply: $P (A and B) = P (A) \times P (B)$ .
OR (mutually exclusive events): add: $P (A or B) = P (A) + P (B)$ .

On a tree diagram, multiply along the branches and add between the branches.

Worked example

A bag has $3$ red and $2$ blue counters. Two are taken without replacement. Find $P (both red)$ .

$P = \frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10} .$ The second fraction changes because a red has been removed.

The expected number of times an event happens in $n$ trials is $n \times P (event)$ , useful for "how many would you expect" questions.

Worked example

A biased coin lands heads with probability $0.3$ . In $200$ throws, how many heads are expected?

$200 \times 0.3 = 60 heads .$

Forgetting that "without replacement" changes the second probability.
Adding when events are independent (should multiply) or multiplying when they are mutually exclusive (should add).
Letting branch probabilities fail to sum to $1$ .

Probability

Overview

Single events

Combined events: AND / OR

Expected outcomes

Common mistakes