13. Fox Alice and Cat Basilio (8th grade. 2 points). Every day, Cat Basilio and Fox Alice visit all 20 courtyards of the capital of the Country of Fools, and in each courtyard, they either receive or do not receive one gold coin with a probability of $\frac{1}{2}$. If the number of gold coins collected by the end of the day is even, they split them equally. If it is odd, they split all but one coin equally, and the last coin is taken by Cat Basilio as the commercial director of the enterprise. Find the expected number of coins received by the cat per day.
Solution: Let $X$ be the revenue of the cat, and $Y$ be the revenue of the fox. Then the random variable $X+Y$ has an expected value of 10. The random variable $X-Y$ with equal probabilities takes the values 0 and 1. Therefore, $\mathrm{E}(X-Y)=0.5$.

 Represent $X$ as $\frac{(X+Y)+(X-Y)}{2}$. We get:

$$
\mathrm{E} X=\frac{\mathrm{E}(X+Y)+\mathrm{E}(X-Y)}{2}=\frac{10+0.5}{2}=5.25.
$$

Answer: $5.25$.