Conditional probability

The idea

$P(A \mid B)$ is the probability of $A$ when you know $B$ has happened: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ . In a table or Venn diagram, the group you are told about becomes the whole.

Find $P(B)$ , the condition: on a tree, add every path that gives $B$ .
Find $P(A \cap B)$ .
Divide.

Watch out The intersection goes on top and the condition on the bottom.

See it worked

Worked example

$P(\text{rain}) = 0.3$ , $P(\text{late} \mid \text{rain}) = 0.4$ and $P(\text{late} \mid \text{no rain}) = 0.1$ . Find $P(\text{late})$ .

Rain and late: $0.3 \times 0.4 = 0.12$ .
No rain and late: $0.7 \times 0.1 = 0.07$ .
$P(\text{late}) = 0.12 + 0.07 = 0.19$ .

Answer $0.19$

Your turn

Question 1Non-calculator3 marks

$P(A \cap B) = 0.12$ and $P(B) = 0.4$ . Find $P(A \mid B)$ .

Question 2Non-calculator3 marks

Of $30$ students, $13$ study physics. Of those, $8$ also study maths. A student who studies physics is chosen at random. Find the probability that they study maths.

Question 3Calculator allowed3 marks

Using $P(\text{rain}) = 0.3$ , $P(\text{late} \mid \text{rain}) = 0.4$ and $P(\text{late} \mid \text{no rain}) = 0.1$ , find $P(\text{rain} \mid \text{late})$ .

In past papers

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