In Anchuria, there are $K$ laws and $N$ ministers. The probability that a randomly chosen minister knows a randomly selected law is $p$. One day, the ministers gathered for a council to write a Concept. If at least one minister knows a law, then this law will be taken into account in the Concept; otherwise, this law will not be included in the Concept. Find:

a) The probability that exactly $M$ laws will be taken into account in the Concept.

b) The expected number of laws taken into account.
a) The probability that no minister knows the first law is $(1-p)^{N}$. Therefore, the probability that the first law will be taken into account is

$q=1-(1-p)^{N}$.

The probability that exactly $M$ laws out of the $K$ available will be considered in the Concept can be easily found using the binomial probability formula:

$$
\mathrm{P}_{M}=C_{K}^{M} q^{M}(1-q)^{K-M}=C_{K}^{M}\left(1-(1-p)^{N}\right)^{M}(1-p)^{N \cdot(K-M)}
$$

b) Let's introduce an indicator - a random variable $I_{k}$, which is equal to 1 if the law numbered $k$ is considered and 0 otherwise. $\mathrm{E} I_{k}=1-(1-p)^{N}$.

The number of considered laws $X$ is equal to the sum of these variables: $X=I_{1}+I_{2}+\ldots+I_{\mathrm{K}}$. Therefore, $\mathrm{E} X=\mathrm{E} I_{1}+\mathrm{E} I_{2}+\ldots+$ $\mathrm{E}_{K}=K\left(1-(1-p)^{N}\right)$

## Answer

a) $C_{K}^{M}\left(1-(1-p)^{N}\right)^{M}(1-p)^{N \cdot(K-M)}$; b) $K\left(1-(1-p)^{N}\right)$.

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