What is a conditional probability for 3 dice?

Question

I roll three dice. Let's designate $A$, $B$, $C$ - the number of points on the first, second and third dice, respectively. I need to find the probability that points $A + B > C$ under condition $A <B <C$.

My attempt. I found the possible outcomes $6^3=216$ and I wrote out seven possibilities $(2,3,4), (2,4,5), (2,5,6), (3,4,5), (3,4,6), (3,5,6), (4,5,6)$.

I get the probability 7/216.

Question. How to find the probability through the conditional probability forumula or the product of probabilities?

Answer 1

Let F be the event that $A+B>C$. Let G be the event that $A<B<C$.

You've already found $P(F \cap G) = \frac{7}{216}$

We want $P(F|G)$ which is the probability of F given G.

$P(F|G) = \frac{P(F \cap G)}{P(G)}$.

So we need $P(G)$.

If we choose 3 distinct numbers out of 6, and arrange them from least to greatest, then we have A<B<C. So the number of possibilities is $6 \choose 3$ = $20$.

So $P(G) = \frac{20}{216}$

So $P(F|G) = \frac{\frac{7}{216}}{\frac{20}{216}} = \frac{7}{20}$

EDIT: Or rather than using the conditional probability formula, you can just directly say the answer is $\frac{7}{20}$ because you have 7 good possibilities out of a total of 20.