Task 7. Among some products, the share of top-grade products is $31 \%$. What is the most probable number of top-grade products in a randomly selected batch of 75 products?
Solution. It is known that $p=0.31; q=1-p=0.69; n=75$. By formula (22) we have: $75 \cdot 0.31-0.69 \leqslant k_{0} \leqslant 75 \cdot 0.31+0.31 ; 22.56 \leqslant$ $\leqslant k_{0} \leqslant 23.56 ;$ since $k_{0}$ must be an integer, it follows that $k_{0}=23$.

Answer. 23.

## Exercises

1. Two equally skilled chess players play chess (draws are not considered). Find the probabilities of winning: 3 games out of 6; no more than 3 games out of 6; more than 3 games out of 6.
2. The probability of event $A$ in any of 4 independent trials is 0.8. Find the most probable number of occurrences of event $A$ and indicate the probability of this number.
3. The probability that a part is standard is 0.75. A batch of 10 parts is inspected. Find the most probable number of parts that will be considered standard.
4. How many trials need to be conducted so that the most probable number of occurrences of event $A$ in these trials is 30, if $p=p(A)=0.3$?

## Answers

1. $p_{6}(3)=0.3125 ; p_{6}(k \leqslant 3)=0.65625 ; p_{6}(k>3)=0.34375$. 2. $k_{0}=3$ or $k_{0}=4, p_{4}(3)=p_{4}(4)=0.4096$. 3. 8 4. From 99 to 102. Hint. Apply formula (22).

## § 8. Poisson's Formula. Event Flow

$1^{\circ}$. If the number of trials $n$ is large, and the probability of the occurrence of event $A$ in each trial is small ( $p<0.01$ ), then the numerical implementation of the Bernoulli formula becomes complex, although the formula is valid for any $n$ and $p$. In such cases, Poisson's theorem can be used.

Theorem 8 (Poisson). If $0<p<1$, then

$$
p_{n}(k)=\frac{\lambda^{k} \cdot e^{-\lambda}}{k!} \quad(\lambda=n p ; k=0,1, \ldots, n)
$$

This formula belongs to Poisson and is derived from the Bernoulli formula by taking the limit as $n \rightarrow+\infty$.

$2^{\circ}$. Poisson's formula is applicable for calculating the probabilities of events that form an event flow - a sequence of events that occur at random moments in time.

The intensity of the flow $\lambda$ is the average number of events that occur per unit of time.

A simple, or Poisson, flow of events is one for which the probability of the occurrence of $k$ events in the flow with intensity $\lambda$ is calculated using Poisson's formula

$$
\begin{aligned}
p_{t}(k)= & \frac{(\lambda t)^{k} \cdot e^{-\lambda t}}{k!}, \quad k=0,1,2, \ldots, n . \\
& \text { Problems with solutions }
\end{aligned}
$$