5. If four dice are thrown at random, the probability that the smallest number shown by these four dice is exactly 3 is
$\qquad$ (Supplied by Zhao Bin)
5. $\frac{175}{1296}$.

Let "the minimum number shown on four dice is exactly 3" be event $A_{3}$, and "the minimum number shown on four dice is not less than $i$" be event $B_{i}(i=1,2,3,4,5,6)$, then $A_{3}=B_{3} \backslash B_{4}$. Also,
$$
P\left(B_{i}\right)=\frac{\left|B_{i}\right|}{|\Omega|}=\frac{(7-i)^{4}}{6^{4}}, \quad i=1,2,3,4,5,6,
$$

Therefore,
$$
P\left(A_{3}\right)=P\left(B_{3}\right)-P\left(B_{4}\right)=\left(\frac{4}{6}\right)^{4}-\left(\frac{3}{6}\right)^{4}=\frac{175}{1296} .
$$