13. Test (from 8th grade. 3 points) Vasya and Misha were taking a test, which had 20 questions of the same difficulty, and for each question, answer options were provided. Since neither Vasya nor Misha knew anything, they only guessed the answers. Vasya guessed the correct answers to 6 questions, and Misha to 8 questions, without consulting each other. Find the mathematical expectation of the number of coincidences, that is, the questions where both Vasya and Misha either guessed the correct answers or did not guess them.
Solution. Let the random variable $I_{k}$ be the indicator of a match in the $k-\mathrm{th}$ question, that is, $I_{k}=1$ if both Vasya and Misha either guessed or did not guess the correct answer to question $k$, and $I_{k}=0$ if one guessed and the other did not.

According to the problem, Vasya will guess the correct answer to this question with a probability of $\frac{6}{20}=0.3$, and Misha with a probability of $\frac{8}{20}=0.4$. They will not guess the answer with probabilities of 0.7 and 0.6, respectively. Since Vasya and Misha work independently of each other, the probability that they both guess or both do not answer this question is

$$
0.3 \cdot 0.4 + 0.7 \cdot 0.6 = 0.54
$$

We obtain the distribution

$$
I_{k} \sim\left(\begin{array}{cc}
0 & 1 \\
0.46 & 0.54
\end{array}\right)
$$

The expected value is $\mathrm{E} I_{k}=0.54$. The random variable $X$ "Number of matches" is equal to the sum of the indicators of matches for all questions:

$$
X=I_{1}+I_{2}+\ldots+I_{20}
$$

We transition to expectations:

$$
\mathrm{E} X=20 \mathrm{E} I_{1}=20 \cdot 0.54=10.8
$$

Answer: $10.8$