A and B are independent witnesses in a case, the chance that A speaks

1.	A and B are independent witnesses in a case, the chance that A speaks truth is x and B speaks truth is y, If A and B agree on certain statements, the probability that the statement is true is1. \(\rm \dfrac{xy}{xy+(1-x)(1-y)}\)2. \(\rm \dfrac{xy}{(1-x)(1-y)}\)3. \(\rm \dfrac{(1-x)(1-y)}{xy+(1-x)(1-y)}\)4. \(\rm \dfrac{x+y}{xy+(1-x)(1-y)}\)
Answer» Correct Answer - Option 1 : \(\rm \dfrac{xy}{xy+(1-x)(1-y)}\) Concept: Let A1, A2, …. , An be n mutually exclusive and exhaustive events of the sample space S and A is event which can occur with any of the events then \({\rm{P}}\left( {\frac{{{{\rm{A}}_{\rm{i}}}}}{{\rm{A}}}} \right) = \frac{{{\rm{P}}\left( {{{\rm{A}}_{\rm{i}}}} \right){\rm{P}}\left( {\frac{{\rm{A}}}{{{{\rm{A}}_{\rm{i}}}}}} \right)}}{{\mathop \sum \nolimits_{{\rm{i}} = 1}^{\rm{n}} {\rm{P}}\left( {{{\rm{A}}_{\rm{i}}}} \right){\rm{P}}\left( {\frac{{\rm{A}}}{{{{\rm{A}}_{\rm{i}}}}}} \right)}}\) Calculations: Consider, Let K be the event that both A and B agree, T be the event that they both A and B speak the truth ⇒ P(T) = xy L be the event that they both A and B lie. ⇒ P(L) = (1 -x)(1 - y) To find :The probability that the statement is true = \(\rm P(\frac T L)\) Let K be the event that both of them agree \(\rm P(\frac T L)\) = \(\rm \dfrac {P(T)P(\frac K T)}{P(T)P(\frac {K}{T})+P(L)P(\frac {K}{L}) }\) ⇒\(\rm P(\frac T L)\) = \(\rm \dfrac{xy}{xy+(1-x)(1-y)}\)

A and B are independent witnesses in a case, the chance that A speaks truth is x and B speaks truth is y, If A and B agree on certain statements, the probability that the statement is true is1. \(\rm \dfrac{xy}{xy+(1-x)(1-y)}\)2. \(\rm \dfrac{xy}{(1-x)(1-y)}\)3. \(\rm \dfrac{(1-x)(1-y)}{xy+(1-x)(1-y)}\)4. \(\rm \dfrac{x+y}{xy+(1-x)(1-y)}\)

Answer» Correct Answer - Option 1 : \(\rm \dfrac{xy}{xy+(1-x)(1-y)}\)

Concept:

Let A1, A2, …. , An be n mutually exclusive and exhaustive events of the sample space S and A is event which can occur with any of the events then

\({\rm{P}}\left( {\frac{{{{\rm{A}}_{\rm{i}}}}}{{\rm{A}}}} \right) = \frac{{{\rm{P}}\left( {{{\rm{A}}_{\rm{i}}}} \right){\rm{P}}\left( {\frac{{\rm{A}}}{{{{\rm{A}}_{\rm{i}}}}}} \right)}}{{\mathop \sum \nolimits_{{\rm{i}} = 1}^{\rm{n}} {\rm{P}}\left( {{{\rm{A}}_{\rm{i}}}} \right){\rm{P}}\left( {\frac{{\rm{A}}}{{{{\rm{A}}_{\rm{i}}}}}} \right)}}\)

Calculations:

Consider, Let K be the event that both A and B agree,

T be the event that they both A and B speak the truth

⇒ P(T) = xy

L be the event that they both A and B lie.
⇒ P(L) = (1 -x)(1 - y)

To find :The probability that the statement is true = \(\rm P(\frac T L)\)

Let K be the event that both of them agree

\(\rm P(\frac T L)\) = \(\rm \dfrac {P(T)P(\frac K T)}{P(T)P(\frac {K}{T})+P(L)P(\frac {K}{L}) }\)

⇒\(\rm P(\frac T L)\) = \(\rm \dfrac{xy}{xy+(1-x)(1-y)}\)