TITLE: Rolling $4$ dice and multiplying the results. What is the probability that the product is divisible by $5$ or has $5$ as the least significant digit?
QUESTION [5 upvotes]: Four fair dice are rolled and the four numbers shown are multiplied together. What is the probability that this product
(a) Is divisible by $5$?
(b) Has last digit $5$?
MY ATTEMPT
a) A number is divisible by $5$ if and only if it has at least one factor equal to $5$. Let us denote by $F$ the event "it occurs at least one number five". Hence the sought probability is given by
\begin{align*}
\textbf{P}(F) = 1 - \textbf{P}(F^{c}) = 1 - \frac{5^{4}}{6^{4}} = 1 - \left(\frac{5}{6}\right)^{4}
\end{align*}
b) Here is the problem. I am not able to describe properly the target results.
Am I on the right track? Can someone please help me to solve it? Any help is appreciated.

REPLY [8 votes]: As others have noted, for part (b) you can first show that:

the product has last digit $5$ if and only if all the numbers are odd and there is at least one $5$.

From there, I would proceed like this (just slightly simpler than pwerth and José Carlos Santos's approaches):
\begin{align*}
P(\text{all odd AND at least one 5})
&= P(\text{all odd}) - P(\text{all odd and no 5}).
\end{align*}
So, what is the probability that all dice are odd? And what is the probability all are odd and there is no $5$ (i.e., all dice are $1$ or $3$)?