Probability that the statement is true A speaks truth $3$ times out of $4$ and $B$ $7$ times out of $10$ . they both agree that a white ball has been drawn out from a bag containing $6$ balls of different color . find the probability that the statement is true . 
my try .
probability when they say false and agree = $\left(\dfrac56\right)\left(\dfrac14\right)\left(\dfrac3{10}\right)$
probability when they say correct = $\left(\dfrac34\right)\left(\dfrac7{10}\right)\left(\dfrac16\right)$
but from this I am not getting the answer . 
the answer gibpven as $\dfrac{35}{36}$
 A: Correct Problem
Setting

There is a bag with six balls in it. Each ball has a different color. One ball is blindly drawn. Two people are being asked which color the ball has after they saw the ball.
Person $A$ tells to truth in $3$ out of $4$ cases. Person $B$ tells the truth in $7$ out of $10$ cases. If one of them doesn't tell to truth the wrong answers are uniformly distributed. Which means nothing else than
$$P(A \text{ says "blue" }|\;A \text{ lies}) = \frac{1}{5} \quad \text{if "blue" isn't the correct answer}$$ 
the same counts for $B$.
Question

What is the Probability that the drawn ball is white, if person $A$ and person $B$ both are telling that it is white?
Solution

The actual experiment of drawing a ball doesn't matter any more. Only the distribution of the wrong answers matter.
So we need to calculate the probability
$$P(A \text{ tells the truth } | \; A = B) = \frac{P(A\text{ tells the truth } \cap A=B  )}{P(A=B)}$$
$A = B$ means both give the same answer.
case 1:
both are telling the truth: $\frac{3}{4}\frac{7}{10}$
case 2:
both are lying the same answer: $\frac{1}{4}\frac{3}{10}\frac{1}{5}$
Altogether we receive: $$P(A=B) = \frac{3}{4}\frac{7}{10}+\frac{1}{4}\frac{3}{10}\frac{1}{5} = \frac{108}{200}$$
Now we also know
$$P(A\text{ tells the truth } \cap A=B  ) = P(A \text{ and } B \text{ tell the truth}) = \frac{3}{4}\frac{7}{10} = \frac{21}{40}$$
The last step is building the quotient.
$$\frac{\frac{21}{40}}{\frac{108}{200}} = \frac{35}{36}$$
A: Ans: $\dfrac{35}{36}$
From the problem, 
P(A speaking truth) $=\dfrac{3}{4}$
P(A not speaking truth) $=1-\dfrac{3}{4}=\dfrac{1}{4}$
P(B speaking truth) $=\dfrac{7}{10}$
P(B not speaking truth) $=1-\dfrac{7}{10}=\dfrac{3}{10}$
P(drawing a white ball) $=\dfrac{1}{6}$
P(drawing a non-white ball) =$\dfrac{5}{6}$
Let $X$ be the event that a white ball is drawn and both assert that it is white(in this event, both speak truth).
$P(X)=\dfrac{1}{6}×\dfrac{3}{4}×\dfrac{7}{10}=\dfrac{7}{80}$
Let $Y$ be the event that a non-white ball is drawn and both assert that it is white (in this event, both lie and say it is white).
Finding probability for this event is tricky. To explain this better, assume that the colors of the balls are red, green, blue, black, yellow and white.
Suppose a red ball is taken.  Then there is a probability of $\dfrac{1}{4}$ that A lie and say it is green or blue or black or yellow or white. Therefore, for A to say, it is a white ball, probability $=\dfrac{1}{4}×\dfrac{1}{5}=\dfrac{1}{20}$
i.e., if a non-white ball is taken, probability that A say it is white 
$=\dfrac{1}{4}×\dfrac{1}{5}=\dfrac{1}{20}$
Similarly, if a non-white ball is taken, probability that B say it is white 
$=\dfrac{3}{10}×\dfrac{1}{5}=\dfrac{3}{50}$
Coming back to the event $Y$ where a non-white ball is drawn and both assert that it is white.
$P(Y)=\dfrac{5}{6}×\dfrac{1}{20}×\dfrac{3}{50}=\dfrac{1}{400}$
$X$ and $Y$ covers all the cases where A and B assert that the ball drawn is white. Note that assertion is true for event $X$ and false for event $Y$
Hence, required probability
$=\dfrac{\dfrac{7}{80}}{\dfrac{7}{80}+\dfrac{1}{400}}=\dfrac{35}{35+1}=\dfrac{35}{36}$
A: if a ball is drawn and they are asked if it is white true/false and they both say true - then 
P(White | say true) = P(white and say it's true) / P(say true) = 
(1/6)(7/10)(3/4) / ((1/6)(7/10)(3/4) + (5/6)(3/10)(1/4)) 
= (7/80) (7/80 + 1 / 16) = 7 / 12
one in 16 times  - it isn't white, but they lie and say true to the statement it isn't white.  7 in 80 times it is white and they untruthfully agree it is white.  Therefore it seems impossible that the answer is 35/36 - that's why I over-thought it a bit in the comments
