Mutually exclusive events

Two events are said to be mutually exclusive, if they cannot both occur simultaneously.

The Venn diagram below represents the situation in which set A and B are mutually exclusive or disjoint

in this case p(A) = p(B) = p(AuB)

Addition Law

If events A, B,C … are mutually exclusive the probability of A or B or C or …. happening is the sum of their individual probabilities

i.e P(A)  f p (b) +P (C) + ……………………..

Complementary events

Recall that E is the complement of the set

pr (E^c) = 1 – pr(E)

Ex. 1 A number is chosen at random from the set {2,4,6, … 20). Find the probability that it is either a factor of 18 or a multiple of S.

Solution

n(S) = 10

n(factor of 18) = 2,3,6,18,9

= 2,6,18

Therefore = n(factors of 18) =3

p( factor of 18) = 3/10

Multiple of  5 =5,10,15,20

n(multiple of 5) =2

in the set

p(multiple of 5) = 2/10

Independent Events

Two events are said to be independent if the two events have no effect on each other . For example the task of getting both a six and a tail in a throw

Ex 2: Five girls and three boys put their names in a box. One name is picked out at random without replacing the first name; a second name is picked out at random. What it the probability that both are names of girls?

Solution

p (picking a girl) = 5/8

P(picking another girl without replacement) = 4/7

pr (both are names of girls) =

Outcome tables , tree diagrams

Ex 4 A bag contains three black balls and two white ball. A ball is taken from the bag and then replace. A second ball is choose.

What is the probability that

1. a) they are both white.
2. b) one is black and one is white
3. c) at least one is black
4. d) at most is black

Solution

The possible ways of selecting the balls are shown on a tree diagram