Conditional probability

By the end of this lesson

By the end you can calculate $P(A \mid B)$ , the probability of $A$ given that $B$ has happened, both by shrinking the sample space and by using the formula $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ .

Extra information changes the question

Conditional probability answers a question asked after some information arrives.

A standard pack of cards has $52$ cards in four suits (hearts, diamonds, clubs, spades) of $13$ cards each: ace, $2$ to $10$ , jack, queen and king. The jack, queen and king are the picture cards, so there are $3 \times 4 = 12$ of them, and there are $4$ kings.

"A card is drawn from a pack. What is the probability it is a king?" The answer is $\frac{4}{52}$ .

"A card is drawn and you are told it is a picture card. What is the probability it is a king?" Now the pack of $52$ is irrelevant, because we already know the card is one of the $12$ picture cards. The answer is $\frac{4}{12} = \frac{1}{3}$ .

The notation $P(A \mid B)$ means "the probability of $A$ , given $B$ ". The vertical bar is read as "given".

Definition

Conditional probability

For events $A$ and $B$ with $P(B) > 0$ (where $A \cap B$ means " $A$ and $B$ both happen"),

$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$

The condition $P(B) > 0$ matters: you cannot condition on an event that never happens.

The sample space shrinks

When the outcomes are equally likely, the formula is just the counting formula applied to a smaller world.

Knowing $B$ happened throws away every outcome outside $B$ . The new sample space is $B$ itself, and inside it, the outcomes that also belong to $A$ are exactly $A \cap B$ .

$P(A \mid B) = \frac{\text{outcomes in } A \cap B}{\text{outcomes in } B}$

Dividing the top and bottom of that fraction by the size of the original sample space turns the counts into probabilities, which gives the formula above. Both versions always agree — use whichever the question makes easier.

Worked example: Example 1: shrinking the sample space

A fair six-sided die is rolled. You are told the score is greater than $3$ . Find the probability that the score is even.

Have a go first?

Check 1 — a reduced sample space

CalculateQuestion 1 · Apply

CalculateQuestion 2 · Apply

Answer the 2 questions above to continue.