What is the probability when rolling 3 dice that

a) exactly one die shows a 6?

b) at least one die shows a 6?

c) at most one die shows a 6?

(For the example, see the article “>> A” Elements of Probability Calculation <<” in this issue.)
a) The number of possible cases: the number of third-class repetitive variations formed from 6 elements, that is,

$$
V_{6}^{i, 3}=6^{3}=216
$$

The favorable cases: one of the dice shows a 6, and the other two do not. The number of cases where the 6 is on a specified die is clearly the same as the number of second-class repetitive variations that can be formed from 5 elements, that is,

$$
V_{5}^{i, 2}=5^{2}=25
$$

Since the 6 can be on any of the 3 dice, the total number of favorable cases is $3 \cdot 25=75$, and thus the desired probability is

$$
V_{a}=\frac{75}{216}=\frac{25}{72}(\approx 0.347)
$$

b) A 6 does not appear on one die in $V_{5}^{i, 3}=5^{3}=125$ cases, so in $216-125=91$ cases, at least one die shows a 6. Therefore, the desired probability is

$$
V_{b}=\frac{91}{216}(\approx 0.421)
$$

c) At most one 6 means that either $\alpha$) no die shows a 6, or $\beta$) only one die shows a 6.

In the $\alpha$) case, the probability is $\frac{125}{216}$ according to part b), and in the $\beta$) case, the probability is $\frac{75}{216}$ according to part a). Since these two cases are mutually exclusive, the desired probability is

$$
V_{c}=\frac{125}{216}+\frac{75}{216}=\frac{200}{216}=\frac{25}{27}(\approx 0.926)
$$

Without solving parts a) and b), the simplest way to proceed is to calculate the number of unfavorable cases, which are as follows:

I. Two dice show a 6, and the third does not.

II. All three dice show a 6.

In case I, it is possible in 5 ways for a specified die not to show a 6, and since any of the 3 dice can be the one that does not show a 6, the total number of possible cases in this case is $3 \cdot 5=15$.

Case II can only occur in one way.

Thus, the number of unfavorable cases is $15+1=16$, and therefore the number of favorable cases is $216-16=200$ etc.

Alpár Bujdosó (Bp., II., Rákóczi g. III. o. t.)