Two 6-sided die

Let $X,Y$ be the outcomes of rolling two iid six-sided dies. Compute $\mathrm{Pr}(X+Y < XY)$.

Solution

Let $E$ be the event that $X+Y < XY$. If $X=1$, $E$ cannot happen as we $XY = Y$. If $X=2$, then $E$ happens when $Y > 2$. Similarly, if $X>2$ then we need $Y > 2$. Taking into account the symmetry of the problem, the probability is then

\(\begin{gathered} 1 - \mathrm{Pr}(X=1 \text{ or } Y=1) - \mathrm{Pr}(X=Y=2) \\ = 1 - 1/6 - 1/6 + 1/36 - 1/36 = 2/3 \end{gathered}\)