Independent and exclusive events

The idea

Mutually exclusive events cannot happen together: $P(A \cup B) = P(A) + P(B)$ . Independent events do not affect each other: $P(A \cap B) = P(A) \times P(B)$ . In general: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Draw a Venn diagram or a sample space to see every outcome.
Test independence by checking whether $P(A \cap B) = P(A)P(B)$ .
Use the addition rule for 'or'.

Watch out Two events with non-zero probabilities cannot be both mutually exclusive and independent.

See it worked

Worked example

$P(A) = 0.4$ , $P(B) = 0.5$ and $P(A \cap B) = 0.2$ . Are $A$ and $B$ independent?

$P(A) \times P(B) = 0.4 \times 0.5 = 0.2$ .
This equals $P(A \cap B)$ .
So $A$ and $B$ are independent.

Answer Yes, independent

Your turn

Question 1Non-calculator3 marks

$A$ and $B$ are independent, with $P(A) = 0.3$ and $P(B) = 0.6$ . Find $P(A \cup B)$ .

Question 2Non-calculator3 marks

$A$ and $B$ are mutually exclusive, with $P(A) = 0.25$ and $P(B) = 0.4$ . Find $P(A \cup B)$ .

Question 3Non-calculator3 marks

Two fair six-sided dice are rolled. Find the probability that the total is $8$ .

In past papers

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