Independent & Mutually Exclusive Events (DP IB Analysis & Approaches (AA): HL): Revision Note

Independent & mutually exclusive events

What are mutually exclusive events?

• Two events are mutually exclusive if they cannot both occur

  • For example: when rolling a dice the events "getting a prime number" and "getting a 6" are mutually exclusive

• If A and B are mutually exclusive events then:

  • $\mathrm{P}(A\cap B)=0$

What are independent events?

• Two events are independent if one occurring does not affect the probability of the other occurring

  • For example: when flipping a coin twice the events “getting a tails on the first flip” and “getting a tails on the second flip” are independent

• If A and B are independent events then

  • $\mathrm{P}\left(A\right|B)=\mathrm{P}(A)$ and $\mathrm{P}\left(B\right|A)=\mathrm{P}(B)$

    • That is just the maths way of saying 'one occurring does not affect the probability of the other occurring'!

• If A and B are independent events then

  • $\mathrm{P}(A\cap B)=\mathrm{P}\left(A\right)\mathrm{P}\left(B\right)$ 

    • This is given in the exam formula booklet

    • This is a useful formula to test whether two events are independent

How do I find the probability of combined mutually exclusive events?

• If A and B are mutually exclusive events then

  • $\mathrm{P}\left(A\cup B\right)=\mathrm{P}\left(A\right)+\mathrm{P}\left(B\right)$

    • This is given in the exam formula booklet

    • This occurs because for mutually exclusive events $\mathrm{P}\left(A\cap B\right)=0$

• For any two events A and B:

  • The events $A\cap B$ and $A\cap B\text{'}$ are mutually exclusive

  • and A is the union of those two events

    • i.e. $A=\left(A\cap B\right)\cup \left(A\cap B\text{'}\right)$

  • Therefore

    • $\mathrm{P}\left(A\right)=\mathrm{P}(A\cap B)+\mathrm{P}(A\cap B\text{'})$

  • This works for any two events A and B