Question #346996

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?

1
Expert's answer
2022-06-06T14:45:22-0400

1.7.

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

a)


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

=0.52+0.210.52(0.21)=0.6208=0.52+0.21-0.52(0.21)=0.6208

b)


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

=10.6208=0.3792=1-0.6208=0.3792

c)


P(ACBC)=P(AC)+P(BC)P(ACBC)P(A^C\cup B^C)=P(A^C)+P(B^C)-P(A^C\cap B^C)

=(10.52)+(10.21)0.3792=0.8908=(1-0.52)+(1-0.21)-0.3792=0.8908




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

=10.1092=0.8908=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(AB)=0.10920,P(A\cap B)=0.1092\not=0, then AA and BB are not disjoint events.


1.8.

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

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


P(X1)=1P(X=0)P(X\ge 1)=1-P(X=0)

=1(320)(0.15)0(0.85)320=1-\dbinom{32}{0}(0.15)^{0}(0.85)^{32-0}

=0.9944867762=0.9944867762


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