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 Video by Zeeshan Zamurred Conditional Probability

Edexcel A level Maths: 2.2 Conditional ProbabilityWatch the full walkthrough 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 $P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$ — the fraction of $B$ ’s outcomes that are also in $A$ .

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

Conditional probability.

\frac{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 $P (A ∣ B) = P (A)$ — knowing $B$ tells you nothing about $A$ . Equivalently $P (A \cap B) = P (A) P (B)$ .

Formula recap

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

Conditional probability.

P (A \cap B) = P (A) P (B) (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).

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