19. Football-2. (from 10th grade. 6 points) A regular football match is in progress. A draw is possible. The waiting time for the next goal does not depend on previous events in the match. It is known that the expected number of goals in football matches between these teams is 2.8. Find the probability that an even number of goals will be scored during the match.
Solution. Under the given conditions, the waiting time for the next goal is a random variable with an exponential distribution, and the total number of goals in the match is distributed according to the Poisson law with parameter $\lambda=2.8$, that is, the probability that $n$ goals will be scored is $p_{n}=\frac{\lambda^{n}}{n!} e^{-\lambda}$.

Therefore, the probabilities of an even and odd number of goals scored are respectively

$$
p=\left(1+\frac{\lambda^{2}}{2!}+\frac{\lambda^{4}}{4!}+\ldots\right) e^{-\lambda} \text { and } q=\left(\lambda+\frac{\lambda^{3}}{3!}+\frac{\lambda^{5}}{5!}+\ldots\right) e^{-\lambda}
$$

their sum is 1, and their difference is

$$
p-q=\left(1-\lambda+\frac{\lambda^{2}}{2!}-\frac{\lambda^{3}}{3!}+\ldots\right) e^{-\lambda}=e^{-\lambda} \cdot e^{-\lambda}=e^{-2 \lambda}
$$

We obtain the system

$$
\left\{\begin{array}{l}
p+q=1 \\
p-q=e^{-2 \lambda},
\end{array} \text { from which } p=\frac{1+e^{-2 \lambda}}{2} \approx 0.502\right.
$$

Answer: approximately 0.502.

Note. The solution to this problem is also the solution to problem 10: the probability of an even number of goals is greater. The only thing missing in problem 10 is the condition that the expected number of goals scored is finite. This fact can be considered obvious, as there has never been a match in history where an infinite number of goals were scored.

[^0]:    ${ }^{1}$ The Fields Medal (angl. Fields Medal) - an international prize and medal that are awarded every 4 years at each International Mathematical Congress to two, three, or four young mathematicians.

[^1]:    ${ }^{2}$ We use the formula for the variance of a set of numbers $S^{2}=\overline{x^{2}}-\bar{x}^{2}$, where $\bar{x}-$ is the arithmetic mean of the numbers, and $\overline{x^{2}}-$ is the arithmetic mean of their squares.

[^2]:    ${ }^{3}$ For example, the sequence 00011011100 consists of five groups $000,11,0,111$ and 00.