TITLE: Probability question is bugging me
QUESTION [7 upvotes]: I recently saw a question somewhere where I got confused between what exactly I should do about it.
Q. Imagine person A speaks truth 9 out of 10 times and another person B speaks truth 8 out of 10 times. A random card is picked from Jack, Queen and Kings (12 cards total). If both A and B say the random card is Jack of Clubs, what is the probability that the Jack of Clubs was not the picked card?
A. In the answer the questioner said, the answer is supposed to be 1/144 because both are having 12 possibilities of saying something. I thought it was either 2/100 since then both have lied or 1/37  since if both say same card, then either both are lying or both are truthful and hence 2/(2+72).
Please tell me which is the correct answer and also please explain why. I am getting confused because of the questioners answer ignoring the truthfulness of A and B's claim.

REPLY [3 votes]: Since they both affirm it is a JC, then either

the card is actually a JC, i.e. both tell the truth, or
the card is not a JC, i.e. both are lying, and in lying they both select JC.

The space of all possible events corresponds to the following Venn diagram

and is
$$
U = {1 \over {12}}\left( {\left( {{{8 + 2} \over {10}}} \right)\left( {{{9 + 1} \over {10}}} \right)} \right) +
 {{11} \over {12}}\left( {\left( {{{8 + 2} \over {10}}} \right)\left( {{{9 + 1} \over {10}}} \right)} \right) = 1
$$
The space of the event $X$ , corresponding to A and B affirming that the card is JC,
whether it is actually so or not is
$$
P(X) = {1 \over {12}}{9 \over {10}}{8 \over {10}} + {{11} \over {12}}{1 \over {10}}{2 \over {10}}
{1 \over {11^{\,2} }} = {{11 \cdot 9 \cdot 8 + 2} \over {12 \cdot 100 \cdot 11}}
$$
and the sought probability is therefore
$$
P = {{{2 \over {12 \cdot 100 \cdot 11}}} \over {{{11 \cdot 9 \cdot 8 + 2} \over {12 \cdot 100 \cdot 11}}}}
 = {2 \over {794}} = {1 \over {397}}
$$