Conditional Probability - Problem Solving

A lot of difficult probability problems involve conditional probability. These can be tackled using tools like Bayes' Theorem, the principle of inclusion and exclusion, and the notion of independence.

Two standard dice with 6 sides are thrown and the faces are recorded. Given that the sum of the two faces equals to 10, what is the probability that the first throw equals to 5?

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Let $A$ denote the event for which the two faces sums to 10, and $B$ the event for which the first throw equals 5.
Then the sample space for $A$ is $\{ (5,5),(4,6),(6,4) \}$.
The sample space for $B$ is $\{ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6) \}$.

Therefore, $A \cap B = \{(5,5)\}$ and $P(B|A) = \frac{P(A\cap B)}{P(A)} = \frac{1/36}{3/36} = \frac13. \ _\square$