If 2 dice are rolled what is probability that "product of given values of dices is > 10" | "given the result is a double"

Question

I am new in the field of probabilities. I came accross this problem. Is my solution correct?

If 2 dices are rolled, what is the probability of P("product of given values of dices is > 10" | "given the result is a double")?

I think that P("values are identical") = 6/36  # (1,1),(2,2)..(6,6)
And all double dices whose product is greater than 10 is (4,4),(5,5),(6,6)
P("product of given values of dices is > 10" | "values are identical") = (3/6) / (6/36)

Is this true? Thank you.

Answer 1

You can calculate the conditional probability of the events like you did:

Let $ X = \text{product of given values of dices is > 10}$ and $Y = \text{values are identical}$ $$P(X| Y) = \frac{P(X \land Y)}{P(Y)} = \frac{\frac{3}{36}}{\frac{6}{36}}$$

Your problem was taking the sample space of $P(X \land Y)$ to be the set of identical throws, where it is the whole set of two dice throws $\Omega$