78. Find the distribution series of the random variable $X$ - the number of times 6 points appear when rolling a die once.
Solution. The random variable $X$ is the number of times a 6 is rolled in a single throw of a fair die. As a result of the experiment, it can take one of the following two possible values: $x_{1}=0 ; x_{2}=1$.

Events $A_{1}, A_{2}, \ldots, A_{6}$ - rolling, respectively, 1, 2, ..., 6 points in a single throw of a die are equally probable. The sum of the probabilities of these events is one, therefore

$$
P\left(A_{1}\right)=P\left(A_{2}\right)=\ldots=P\left(A_{6}\right)=1 / 6
$$

Consequently, $p_{2}=P(X=1)=1 / 6$. Event $\bar{A}_{2}$ - not rolling a 6 in a single throw of a die is the opposite of event $A_{2}$, the probability $P\left(\bar{A}_{2}\right)=1-P\left(A_{2}\right)=1-1 / 6=5 / 6$.

The probability $p_{1}=P(X=0)=P\left(\bar{A}_{2}\right)=5 / 6$.

The random variable $X$ has the following distribution:

$$
X: \begin{array}{ccc}
x_{i} & 0 & 1 \\
p_{i} & 5 / 6 & 1 / 6
\end{array}
$$