Answer to Question #346996 in Statistics and Probability for Nzuzi

Question

Question 1.7 [2, 2, 2, 2]

If A and B are independent events with P A( ) 0.52 = and P B( ) 0.21 = , find the following:

a) PA B ( ) ∪

b) PA B ( ) ∩

c) PA B ( ) ∪

d) Are A and B disjoint events? Motivate your answer!

Question 1.8 [3]

A certain washing machine factory has found that 15% of its washing machines manufactured in the

factory break down and are returned in the first year of operation. Suppose that 32 machines are

purchased by a laundromat from this washing machine factory, find the probability that at least one

washing machine breaks down in the first year of operation?

Answer 1

1.7.

$$P(A\cap B)=P(A)P(B)=0.52(0.21)=0.1092$$

a)

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

$$=0.52+0.21-0.52(0.21)=0.6208$$

b)

$$P(A^C\cap B^C)=1-P(A\cup B)$$

$$=1-0.6208=0.3792$$

c)

$$P(A^C\cup B^C)=P(A^C)+P(B^C)-P(A^C\cap B^C)$$

$$=(1-0.52)+(1-0.21)-0.3792=0.8908$$

$$P(A^C\cup B^C)=1-P(A\cap B)$$

$$=1-0.1092=0.8908$$

d)

If two events are disjoint, then the probability of them both occurring at the same time is 0.

Since $P(A\cap B)=0.1092\not=0,$ then $A$ and $B$ are not disjoint events.

1.8.

Let $X=$ the number of washing machine which breaks down in the first year of operation: $X\sim Bin (n, p).$

Given $n=32,p=0.15, q=1-p=0.85.$

$$P(X\ge 1)=1-P(X=0)$$

$$=1-\dbinom{32}{0}(0.15)^{0}(0.85)^{32-0}$$

$$=0.9944867762$$