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 :
[d] 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 \[P({{E}_{1}}/E)=\frac{P({{E}_{1}})P(E/{{E}_{1}})}{P({{E}_{1}})P(E/{{E}_{1}})+P({{E}_{2}})P(E/{{E}_{2}})}\] \[=\frac{xy.1}{xy.1+(1-x)(1-x).1}=\frac{xy}{1-x-y+2xy}\]
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