3.5 Mutually Exclusive Events

Consider two events A and B represented in the Venn diagram below

We know the probability of A or B can be found as:

P(A or B) = n(A or B)/n(S)

We know from the principle of inclusion /exclusion that

n(A or B) = n(A) + n(B) – n(A and B)

so

P(A or B) = (n(A) + n(B) – n(A and B)) /n(S)

= n(A)/n(S) + n(B)/n(S) – n(A and B)/n(S)

= P(A) + P(B) – P(A and B)

But what if A and B = Ø (the empty set) as shown in the diagram below

Then P(A and B) = 0, and our formula above still makes sense:

P(A or B) = P(A) + P(B) – 0

= P(A) + P(B)

Definition: Two or more events that cannot occur at the same time are called mutually exclusive or disjoint events. This is the case where P(A and B) = 0.

Definition: Events that can occur at the same time are called non-mutually exclusive events. This is the case where P(A and B)≠0.

Ex 1 Picking a coloured marble from a bag containing 4 blue marbles, 3 red marbles and 7 green marbles, what is the probability of picking and blue marble or a red marble? (Assume all marbles are a single colour.)

Ex 2: A card is randomly selected from a standard deck of playing cards. What is the probability that either a heart or a face card will be selected?

Homework: pg 340 #1-3, 5-7,11,13,15