Two hockey teams of equal strength agreed to play until the total score reaches 10.

Find the mathematical expectation of the number of times a tie occurred.
If $2 n$ is the maximum total score, the game can be considered as a random walk of length $2 n$: at each step, the score difference either increases or decreases by one.

Let $I_{2 k}$ be the indicator of a tie at the $2 k$-th step: $I_{2 k} \sim\left(\begin{array}{cc}0 & 1 \\ 1-\frac{C_{2 k}^{k}}{2^{2 k}} & \frac{C_{2 k}^{k}}{2^{2 k}}\end{array}\right)$.

The random variable $X$ "the number of times a tie occurred" is the sum of all indicators. The start of the game will not be considered as a "tie occurrence". Therefore, $X=I_{2}+I_{4}+\ldots+I_{2 n}$, and $\quad \mathrm{E} X=\sum_{k=1}^{n} I_{2 k}=\sum_{k=1}^{n} \frac{C_{2 k}^{k}}{2^{2 k}}$.

For $n=5$, we get $\mathrm{E} X=2 / 4+6 / 16+20 / 64+70 / 256+252 / 2048 \approx 1.707$.

## Answer

$\approx 1.707$.

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