A) \[\frac{3}{8}\]
B) \[\frac{1}{5}\]
C) \[\frac{3}{4}\]
D) \[\frac{1}{2}\]
Correct Answer: A
Solution :
[a] P (six occurs)\[=P({{S}_{2}})=\frac{1}{6}\] P (Six does not occur) =\[P({{S}_{2}})=\frac{5}{6}\] \[P(E|{{S}_{1}})=P\] (Man speaks truth) \[=\frac{3}{4}\] \[P\left( E|{{E}_{2}} \right)=P\](Man does not speak the truth)\[=\frac{1}{4}\] \[\therefore \] By Baye?s theorem, \[P({{S}_{1}}|E)=P\] (he reports that six has occurred is actually a six) \[=\frac{P({{S}_{1}})P(E|{{S}_{1}})}{P({{S}_{1}})P(E|{{S}_{1}})+P({{S}_{2}})P(E|{{S}_{2}})}\] \[=\frac{\frac{1}{6}\times \frac{3}{4}}{\frac{1}{6}\times \frac{3}{4}\times \frac{5}{6}\times \frac{1}{4}}=\frac{3}{8}\]
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