The question:
- Q rolls a fair 6-sided dice and tells you the result.
- Q lies 3/4th of the time.
If Q chooses to lie he will always pick 6, unless he got a 6, in which case he will pick a different number with uniform probability.
Given that Q tells you he got a 6, whats the probability he actually got a 6?
My solution:
If Q is saying the truth (with 0.25 probability), getting a 6 would have 1/6 probability. If Q is lying (0.75 probability), he didn't get a 6 so 0 probability.
Therefore the answer must be 1/4 * 1/6 = 1/24
Where am I wrong ?
He says he got 6. There are two possibilities: he did and he is telling the truth; or he didn't and he is lying.
The prob of the first is $\frac{1}{6}\ \frac{1}{4}=\frac{1}{24}$. The prob of the second is $\frac{5}{6}\ \frac{3}{4}=\frac{15}{24}$.
We know that one of these two is the case, so given that, the prob. that it is the first is $$\frac{\frac{1}{24}}{\frac{1}{24}+\frac{15}{24}}=\frac{1}{16}$$