PROBLEM 9E

Suppose that A, B, and C are mutually independent events and that P(A) = 0.5, P(B) = 0.8, and P(C) = 0.9. Find the probabilities that (a) all three events occur, (b) exactly two of the three events occur, and (c) none of the events occurs.

Step 1 of 5:

Given that events A,B and C are mutually independent.

Also it is given that P(A)=0.5,P(B)=0.8 and P(C)=0.9.

Step 2 of 5:

(a)

Here we have to find the probability that all the three events occur.

Probability that all the three events occur is the probability of the event ABC.

Since the events are mutually independent, the probability of ABC is given by

P(ABC)=P(A)*P(B)*P(C)

=(0.5)*(0.8)*(0.9)

=0.36

Therefore, probability of all the three events occur is 0.36.

Step 3 of 5:

(b)

Here we have to find the probability that exactly two of the events occur.

Probability that exactly two of the three events occur is given by,

P(AB)+P(AC)+P(BC)

where, ,, denotes the non occurrence of the events A,B and C respectively.

We know that P(A)+P()=1.

Therefore,

P()=1-P(A)

=1-0.5

=0.5

P()=1-P(B)

=1-0.8

=0.2

and

P()=1-P(C)

=1-0.9

=0.1