2.2StatisticsStretch

Conditional Probability

Conditional probability is the probability of one event given that another has already happened. It is written P(A | B) and computed by restricting attention to the outcomes where B occurs.

24 min Conditional Probability

Edexcel A level Maths: 2.2 Conditional ProbabilityCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Define P(A | B)
Use the conditional probability formula
Recognise independence
Apply to real contexts

The formula

The probability of $A$ given $B$ is $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ — the fraction of $B$ ’s outcomes that are also in $A$ .

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

Conditional probability.

Example 1Given

P(A \cap B) = 0.2

and

P(B) = 0.5

, find

P(A \mid B)

\dfrac{0.2}{0.5}

= 0.4

Answer

0.4

Tip — Conditioning on B means dividing by P(B) — you’re working within B’s world.

Independence

If $A$ and $B$ are independent then $P(A \mid B) = P(A)$ — knowing $B$ tells you nothing about $A$ . Equivalently $P(A \cap B) = P(A)P(B)$ .

Formula recap

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

Conditional probability.

P(A \cap B) = P(A)P(B) \ (\text{independent})

Independence.

Common mistakes to avoid

Dividing by P(A) instead of P(B) for P(A | B).

Divide by the probability of the given event, P(B).

Assuming events are always independent.

Check P(A∩B) = P(A)P(B) before assuming independence.

Key takeaways

P(A | B) = P(A∩B)/P(B).
Conditioning on B means dividing by P(B).
Independent: P(A | B) = P(A), so P(A∩B) = P(A)P(B).

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

$P(A \mid B)$ equals:
- $\frac{P(A \cap B)}{P(B)}$
- $\frac{P(A \cap B)}{P(A)}$
- $P(A)P(B)$
- $P(A) + P(B)$
Show answer and explanation
Answer: $\frac{P(A \cap B)}{P(B)}$
Divide by the given event.
With $P(A \cap B)=0.2$ , $P(B)=0.5$ , $P(A\mid B)=$
- $0.4$
- $0.1$
- $0.7$
- $0.25$
Show answer and explanation
Answer: $0.4$
0.2/0.5.
If A and B are independent, $P(A \mid B)$ equals:
- $P(A)$
- $P(B)$
- $0$
- $1$
Show answer and explanation
Answer: $P(A)$
B gives no info.
Independence means $P(A \cap B) =$
- $P(A)P(B)$
- $P(A)+P(B)$
- $P(A)-P(B)$
- $0$
Show answer and explanation
Answer: $P(A)P(B)$
Multiply.
P(A | B) restricts attention to outcomes where:
- B occurs
- A occurs
- neither occurs
- both fail
Show answer and explanation
Answer: B occurs
Given B.

Next step

Check your understanding

Work through questions on Conditional Probability. Use the feedback to decide what needs another look.

1 past-paper question · 2 marks

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