97. An urn contains 5 white and 10 black balls. A ball is randomly drawn from the urn. Find the distribution of the random variable $X$ - the number of white balls drawn. Calculate the mathematical expectation, variance, and standard deviation of this random variable.
S o l u t i o n. The experiment consists of drawing one ball. In the experiment, event $A$ - the extraction of a white ball - may or may not occur. The probability of event $A$ occurring is found using formula (1): $P(A)=\frac{m}{n}=\frac{5}{15}=\frac{1}{3}$.

Let $P(A)=p, P(\bar{A})=1-p=q$. Thus, $p=1 / 3$, $q=2 / 3$.

The random variable $X$ - the number of white balls drawn as a result of the experiment - can take one of two possible values: $x_{1}=0$, $x_{2}=1$. The probabilities of these values are $2 / 3$ and $1 / 3$, respectively.

The random variable under consideration is characterized by the following distribution:

$$
X: \begin{array}{ccc}
x_{i} & 0 & 1 \\
p_{i} & 2 / 3 & 1 / 3
\end{array}
$$

The expected value, variance, and standard deviation of the variable $X$ are respectively: $M(X)=p=\frac{1}{3}$, $D(X)=p q=\frac{1}{3} \cdot \frac{2}{3}=\frac{2}{9}, \quad \sigma(X)=\sqrt{D(X)}=\frac{\sqrt{2}}{3}$.