# 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 ?

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$xy/(xy+(1-x)(1-y))$ –  picakhu Nov 28 '11 at 16:15
@picakhu: why wouldn't you put this as an answer? –  Ilya Nov 28 '11 at 16:58
@Ilya, I am too lazy to write it up and it is trivial. –  picakhu Nov 28 '11 at 16:59
@picakhu: How is this trivial? Maybe you're a-naturally talented. –  Gigili Nov 28 '11 at 17:03
@Gigili, please refer to answers below to see why. It should be trivial to everyone not just me I hope. –  picakhu Nov 28 '11 at 19:07

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:

$$\text{Prob}(S_t | A_t \cap B_t) = \frac{xy}{1+2xy-x-y}$$

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$P(A)=x$ 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:

$$\frac{xy}{xy+(1-x)(1-y)}$$.

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No, that's probability that both speak truth. –  sdcvvc Nov 28 '11 at 16:54