How to find the probability of truth? 
A and B are independent witness in a case. The probablity that A
  speaks the truth is 'x' and that of B is 'y'.If A and B agree on a
  certain statement, how to find the probability that the statement is
  true ?

 A: $ P(A)=x $ and $ P(B)=y $,
$A$ and $B$ are independent, so $P(A \cap B)=P(A)P(B)$
Therefore, the probability that both speak the truth will be $P(A \cap B)=P(A)P(B)=xy$.
And then, the probability that they agree on a certain statement is, they both speak the truth or they both tell lie, which would be:
$$xy + (1-x)(1-y) $$
As a result, the probability that the statement is true is:
$$ \frac{xy}{xy+(1-x)(1-y)} $$.
A: Let:
$A_t$ stand for "A says statement is true." and $A_f$ for "A says statement is false" and 
$B_t$ stand for "B says statement is true." and $B_f$ for "B says statement is false" and 
$S_t$ stand for "Statement is true" and $S_f$ for "Statement is false" and 
Then, we know that:
$\text{Prob}(A_t | S_t) = \text{Prob}(A_f | S_f) = x$,  and
$\text{Prob}(A_t | S_f) = \text{Prob}(A_f | S_t) = 1-x$,  and
$\text{Prob}(B_t | S_t) = \text{Prob}(B_f | S_f) = y$,  and
$\text{Prob}(B_t | S_f) = \text{Prob}(B_f | S_t) = 1-y$,  and
We want to know:
$\text{Prob}(S_t | A_t \cap B_t)$ 
Using Bayes theorem, we have:
$$\text{Prob}(S_t | A_t \cap B_t) = \frac{\text{Prob}(A_t \cap B_t |S_t) \text{Prob}(S_t)}{\text{Prob}(A_t \cap B_t)}$$ 
But, we know that,
$\text{Prob}(A_t \cap B_t |S_t) = xy$ and
$\text{Prob}(A_t \cap B_t) = \text{Prob}(A_t \cap B_t |S_t) \text{Prob}(S_t) +  \text{Prob}(A_t \cap B_t |S_f) \text{Prob}(S_f)$
Thus,
$\text{Prob}(A_t \cap B_t) = xy \text{Prob}(S_t) + (1-x)(1-y) (1-\text{Prob}(S_t))$
Simplifying the above, we get:
$\text{Prob}(A_t \cap B_t) = (1+2xy-x-y) \text{Prob}(S_t)$
Thus, we have:
$$\text{Prob}(S_t | A_t \cap B_t) = \frac{xy}{1+2xy-x-y}$$
A: Say the fact is True or False (T/F), and independent statements A, B are binary.
We are given P(A=1|T)=x and P(B=1|T)=y.
By convexity P(A=0|T)=1-x and P(B=0|T)=1-y.
We can surmise P(A=0|F)=x and P(B=0|F)=y.
Also by convexity P(A=1|F)=1-x and P(B=1|F)=1-y.
We want P(T|A=1,B=1).
By Bayes rule P(T|A=1,B=1) = P(A=1,B=1|T)P(T)/P(A=1,B=1).
By independence P(T|A=1,B=1) = P(A=1|T)P(B=1|T)P(T)/(P(A=1)P(B=1)).
Plugging in, the numerator is xyP(T).
The denominator requires P(A=1) = P(A=1|T)P(T) + P(A=1|F)P(F) = xP(T) + (1-x)P(F), same for P(B=1).
So I don’t think we are identified without knowing the marginal probability of the truth P(T).
Of course, for a given x and y you can provide P(T|A=1,B=1) as a function of P(T) between 0 and 1.
