A = event that it is actually a six.
B = event that the girl reports it is a six.
required is P[A|B] i.e. Probability that it is actually a six given that the girl reports so.
So, P[A|B] = {P[B|A].P(A)}/ P(B) .....(1)
P[B|A]= P{Girl reports a six| actually a six}
= P{Girl speaks the truth}
= 4/5
P(A)=P{ it is a six }= 1/6
P(B)=P{Girl reports a six}
=P{Girl reports a six, A}+P{Girl reports a six, it is not a six}
=P[B|A].P(A)+P[B|A'].P(A')
=(4/5).(1/6)+(1/5).(5/6)
= 3/10
Hence, from equation (1)
P[A|B]= {(4/5)(1/6)}/(3/10) = 4/9
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