Independent events and multiplying probabilities

8 minAS Level5.3.25.3.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find $P(A \text{ and } B)$ for independent events, decide whether two events are independent by comparing $P(A \cap B)$ with $P(A) \times P(B)$ , and explain why sampling without replacement usually makes repeated draws dependent.

What "and" means in probability

"A and B" means both happen, written $A \cap B$ (the symbol $\cap$ means "and"). For two stages of an experiment, it means the first stage gives an outcome in $A$ and the second gives an outcome in $B$ .

When the two stages do not influence each other, the probabilities multiply:

$P(A \cap B) = P(A) \times P(B)$

Events for which this multiplication works are called independent. Two coin tosses are independent because the second toss has no memory of the first.

Definition

Independent events

Two events $A$ and $B$ are independent if the occurrence of one does not change the probability of the other.

Equivalently, $P(A \cap B) = P(A) \times P(B)$ .

This equation is also the test for independence: compute both sides from the information given and compare.

Why the test works

Imagine repeating the experiment many times. If $A$ happens in a proportion $P(A)$ of trials, and $B$ happens independently in a proportion $P(B)$ of those trials, then both happen in a proportion $P(A) \times P(B)$ of the whole set.

Independence is about information, not about physical separation. Two events in the same experiment can still be independent if knowing one tells you nothing about the other.

For example, "the first die shows 3" and "the second die shows 5" are independent. But "the first die shows 3" and "the total of two dice is 8" are not: the total being $8$ tells you a good deal about the first die.

Worked example: Example 1: testing whether events are independent

For two events $A$ and $B$ , $P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \cap B) = 0.15$ . Determine whether $A$ and $B$ are independent.

Have a go first?

Check 1 — the independence test

Multiple choiceQuestion 1 · Apply

Answer the 2 questions above to continue.