AS & A Level Maths

Conditional Probability

Conditional probability is the probability of one event given that another event has already happened.

Sometimes we already know that one event has happened, and this changes the sample space.

The probability of event \(B\) given that event \(A\) has happened is written as:

\[ P(B \mid A) \]

This is read as “the probability of \(B\) given \(A\)”.

For any two events \(A\) and \(B\), provided that \(P(A)\ne 0\):

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

This means:
conditional probability = probability of both events happening ÷ probability of the given event.

Rearranging the formula gives:

\[ P(A \cap B)=P(A)\times P(B \mid A) \]

This is useful in tree diagrams and multi-step probability questions.

One card is chosen from a pack of 52 cards.

Let:

There are 12 face cards and 4 kings. Every king is also a face card.

So:

\[ P(B \mid A)=\frac{P(A \cap B)}{P(A)} =\frac{4/52}{12/52} =\frac{1}{3} \]

Therefore, if we already know the card is a face card, the probability that it is a king is \(\frac{1}{3}\).

Read \(P(B \mid A)\) carefully: it means “\(B\) given \(A\)”.
The given event changes the sample space.
Use \(P(B \mid A)=\dfrac{P(A \cap B)}{P(A)}\).
Make sure the event in the denominator is the given event.
Conditional probability is often easier to see using a tree diagram or a Venn diagram.

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