4. Let A and B be events defined on a sample space S with P(A) 0.3 and P(AnB) 0.

Question

4. Let A and B be events defined on a sample space S with P(A) 0.3 and P(AnB) 0.2. (a) Are A and B mutually exclusive? Justify your answer. (b) Is it possible for P(AU B) to be 0? If not what is the smallest value that P(AUB) can take? When P(AUB) takes its smallest value what is the value of P(B) and what is the relationship between A and B? (c) Is it possible for P(AU B) to be 1? If not what is the largestest value that P(AU B) can take? When P(AUB) takes its largest value what is the value of P(B) and what is the relationship between A and B? (d) What are the largest and smallest values that P(B) can take? (e) Find P(B|A) (f) If A and B are independent what is the value of P(B)? [14 marks]

Explanation / Answer

a)
No, if A and B would have been mutually exclusive
then P(A and B) would be 0

b)
No, P(A or B) can not be 0
P(A or B) = P(A) +P(B) - P(A and B)
= 0.3 + P(B) - 0.2
= 0.1 + P(B)
no minimize P(A or B)
we need to minimize P(B)
minimum value of P(B) = 0.2 {P(A and B) }

hence
P(A or B) = 0.3

c) yes, P(A or B) can be 1
when
P(B) = 0.9

d)
min P(B) = 0.2
and max P(B) = 0.9
e) P(B|A) =P(A and B)/P(A)
= 0.2/0.3
=2/3

f)
P(A and B) = P(A) P(B)
hence
P(B) = P(A and B)/P(A) = 2/3

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