There are three fair six-sided dice with sides $0,1,e,\pi,i,\sqrt2$. If these dice are rolled, the probability that the product of all the numbers is real can be expressed as $\frac ab$ where $a$ and $b$ are positive, co-prime integers. What is $a+b$?
When I tried I got the total possibilities to be 216 and the rest I got wrong. Can you help me find the answer to the problem? (I think it is $\frac{99}{216}$).
For the product to be real, either at least one $0$ has to be rolled ($6^3-5^3=216-125=91$ possibilities), or there must be an even number of occurrences of $\mathrm i$, which makes $4^3=64$ possibilities for zero occurrences and $3\cdot4^1=12$ possibilities for two occurrences, for a total of $91+64+12=167$ possibilities. The fraction $167/216$ is already reduced, so the sum is $167+216=383$.