A) \[\frac{x-y}{x+y}\]
B) \[\frac{xy}{1+x+y+xy}\]
C) \[\frac{x-y}{1-x-y+2xy}\]
D) \[\frac{xy}{1-x-y+2xy}\]
Correct Answer: D
Solution :
\[A\] and \[B\] will agree in a certain statement if both speak truth or both tell a lie. We define following events \[{{E}_{1}}=A\]and\[B\]both speak truth\[\Rightarrow P({{E}_{1}})=xy\] \[{{E}_{2}}=A\]and\[B\]both tell a lie\[\Rightarrow P({{E}_{2}})=(1-x)(1-y)\] \[E=A\]and \[B\] agree in a certain statement Clearly,\[P(E/{{E}_{1}})=1\]and\[P(E/{{E}_{2}})=1\] The required probability is \[P({{E}_{1}}/E)\]. Using Baye's theorem \[P({{E}_{1}}/E)=\frac{P({{E}_{1}})P(E/{{E}_{1}})}{P({{E}_{1}})P(E/{{E}_{1}})+P({{E}_{2}})P(E/{{E}_{2}})}\] \[=\frac{xy.1}{xy.1+(1-x)(1-y).1}=\frac{xy}{1-y-y+2xy}\]
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