Consider throwing two six-sided dice. Let X be the sum of the two values and let Y be the product of the two values.

Question

What is the value of $P(X = i, Y = j)$ for $i = 1,2,\cdots,12$ and $j = 1,2,\cdots,36$.

Trying to figure out if there is a easier way to figure this out rather than writing out all the values.

Answer 1

Hint: For $X=2$, there's the one possibility of $(1,1)$. For $X=3$, there's two options $(2,1), (1,2)$. This pattern continues increasing until after $7$, and then the possibilities decrease until $X=12$ with only $(6,6)$.

As for the product $Y$ it's just finding $2$ factors between $1$ and $6$. Note that primes above $6$ logically won't work, and also multiples of those numbers $(11, 22, 33)$.

Answer 2

Consider the system of two equations $i+j=m$, $ij=n$.

By substitution, one gets a quadratic equation. If $i+j=m$, $ij=n$ has a solution with $i$, $j$ integers in the right range, then it either has exactly $2$ solutions or $1$ solution. Thus all probabilities will be $0$, $2k$, or $k$ for some constant $k$. The pairs $(x,y)$ with probability $k$ are $(2,1),(4,4),\dots,(12,36)$.

This speeds up the work considerably, but does not eliminate drudgery entirely.