<s>[INST] A simple conditional probability problem Assume that two fair dice are rolled one at a time. Given that the sum of the two numbers that occured was at least $7$, compute the probability that it was equal to $7$.
I tried computing the probability as follows:
$$P \left( X+Y=7| X+Y \geq 7 \right)=\frac{P \left(X+Y=7 \right)}{P \left(X+Y \geq 7 \right)}$$
And then I examined all different pairs whose sum is $7$ and then those whose sum is greater or equal to $7$. I am getting $3/12$ but the right answer is $6/21$. Could you please help me understand what I am doing wrong? 
Thank you.
 [/INST] $P(X+Y=7) = 1/6$
By Symmetry
$P(X+Y>7) = P(X+Y<7) = (1 - 1/6)/2$
$P(X+Y>=7) = P(X+Y>7)+P(X+Y=7) = 5/12 + 1/6$
$P( X+Y=7| X+Y >= 7) = \frac{1/6}{5/12+1/6} = 6/21$
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