Question

In a sample space, events A and B are independent, events B and C are mutually exclusive, and A and C are independent. a) Show that P(AUB) = P(B) + P(A)P(B') = P(A) + P(A')P(B)

b) If P(AUBUC) = 0.9, P(B) = 0.5 and P(C) = 0.3 find P(A).

Answer 1

a)

as event A and B are independent

P(A n B)=P(A)*P(B)

hence P(A u B)=P(A)+P(B)-P(A)*P(B)

P(A u B)=P(A)+P(B)*(1-P(A))

P(A u B)=P(A)+P(B)*P(A')

similarly P(A u B)=P(B)+P(B')*P(A)

hence P(A u B)=P(B)+P(B')*P(A)=P(A)+P(B)*P(A')

b)as P(A n B)=P(A)*P(B)'

P(B n C)=0

P(A n C)=P(A)*P(C)

for P(A u B u C)=P(A)+P(B)+P(C)-P(A)*P(B)-P(A)*P(C)

0.9=P(A)+0.5+0.3-P(A)*(0.5+0.3)

P(A)*0.2=0.1

P(A)=0.5