Probability : Combined events 1

Objectives

When you have competed it you should

* know the addition rule

* know Venn diagram

* know the mutually exclusive events

Key terms: Addition rule, Mutually exclusive events, Venn diagram, Exhaustive events

Addition Rule

If A and B are any two events of the same experiment then the probability of A or B or both occurring is given by

P ( A or B or both ) = P(A) + P(B) - P(A and B)

P ( A B ) = P(A) + P(B) - P(AB)

ABA B

SVenn diagram

Example 1

Events A and B are such that P(A) = 0.6 , P(B) = 0.7 and P(AB) = 0.4.

P(A B) = P(A) + P(B) - P(A B)

Solutiom

Find P(A or B or both)

A B1 = 0.2 A1 B = 0.3A B = 0.4

P(A B) = 0.6 + 0.7 - 0.4 = 0.9

Mutually Exclusive Events

If an event A can occur or an event B can occur but not both A and B can occur, then the two events A and B are said to be mutually exclusive.

P(AB) = 0

P(A B) = P(A) + P(B)

BA

In a race the probability that Martin wins is 0.3, the probability that Ali wins is 0.25 and the probability that Chun wins is 0.2.

Example 2

Find probability that (a) Martin or Chun wins (b) neither Chun nor Ali wins .

Solution

P( Martin or Chun ) = P(Martin) + P(Chun)

P( Martin or Chun ) = 0.30 + 0.20 = 0.50

P(neither chun nor Ali) = 1 - P( Chun or Ali ) = 1 - ( 0.20 + 0.25 )

= 0.55

Example 3

Solution

Faulty Not Faulty

Machine A

Machine B

8 4

5 3

Tests are carried out on two machines A and B to assess the likelihood that each machine will produce a faulty component.

A component is chosen at random from those tested. Find the probability that the component chosen (i) is from machine A

(ii) is a faulty component from machine B

(i) 12/20 = 3/5 (ii) 5/20 = 1/20