Applying Bayes Theorem

Question

A man speaks the truth $8$ out of $10$ times. A fair die is thrown. The man says that the number on the upper face is $5$. Find the probability that the original number on the upper face is $5$.

While solving I find two ways (shown in the image). I think one of them is correct and other one is incorrect. Please tell me which is the correct one and why.

Any advice on solving tricky problems (on Bayes theorem) is welcome.

Answer 1

Your first one is false. $P(X \mid T)$ is $1$, not $\frac{1}{6}$. Given that the man speaks the truth, the die definitely showed $5$.

Your second method is much more natural, and it is correct (assuming you plugged the numbers in correctly).

Answer 2

Surely the answer is $\frac 8{10}$ as all you are asking is: "is the fellow telling the truth?"

Let's see this following the approach you took.

There are two ways in which one might hear the answer "$5$" from the fellow. Either it is $5$, and he tells the truth about it, or it is not $5$ and he lies and happens to say that it is $5$. Now, here we must make an assumption because the problem doesn't specify how he lies. Absent any information to the contrary, I assume that his lying is "unbiased". that is, when he chooses to lie, he chooses uniformly at random from the available options. If, say, the true value is $4$ then he chooses uniformly at random from $\{1,2,3,5,6\}$. Hence if the die does not show $5$, and he chooses to lie about it, there is a $\frac 15$ probability that he will say $5$ (Note: in your work you appear to assume that when he lies he always says $5$, which seems arbitrary).

The probability of the first path is: $\frac 16\times \frac 8{10}$

The probability of the second path is $\frac 56 \times \frac 2{10} \times \frac 15$

Thus the answer is $$ \frac {\frac 16\times \frac 8{10}}{\frac 16\times \frac 8{10}+\frac 56 \times \frac 2{10} \times \frac 15}=\frac {1\times 8}{1\times 8+2}=\frac 8{10}$$

Note: if you make other assumptions about the lying then you can get other answers. Certainly you'll get a different answer if you assume that his dishonest reply is biased for or against saying $5$, which is what you implicitly assumed.