183. The probability of winning with one lottery ticket is $p=0.01$. How many tickets need to be bought to win at least one of them with a probability $P$, not less than 0.95?
Solution. The probability of winning is small, and the number of tickets that need to be purchased is obviously large, so the number of winning tickets is approximately Poisson distributed.

It is clear that the events "none of the purchased tickets are winning" and "at least one ticket is winning" are opposite. Therefore, the sum of the probabilities of these events is one:

$$
P_{n}(0)+P=1 \text {, or } P=1-P_{n}(0) \text {. }
$$

By setting $k=0$ in the Poisson formula $P_{n}(k)=\lambda^{k} \mathrm{e}^{-\lambda} / k!$, we get

$$
P_{n}(0)=\mathrm{e}^{-\lambda}
$$

Therefore, the relation (*) will take the form

$$
P=1-\mathrm{e}^{-\lambda}
$$

By the condition, $P \geqslant 0.95$, or $1-\mathrm{e}^{-\lambda} \geqslant 0.95$. From this,

$$
\mathrm{e}^{-\lambda} \leq 0.05
$$

Using the table of the function $e^{-x}$, we find $e^{-3}=0.05$. Considering that the function $\mathrm{e}^{-x}$ is decreasing, we conclude that the inequality (**) is satisfied for $\lambda \geqslant 3$, or for $n p \geqslant 3$. Therefore, $n \geqslant 3 / p=$ $=3 / 0.01=300$. Thus, one needs to buy at least 300 tickets to win at least one of them.

## § 2. Simplest Event Flow

An event flow is a sequence of events that occur at random moments in time.

A simplest (Poisson) event flow is one that has the following three properties: stationarity, "lack of aftereffect," and ordinarity.

The property of stationarity means that the probability of $k$ events occurring in any time interval depends only on the number $k$ and the duration $t$ of the time interval and does not depend on the start of its measurement. In other words, the probability of $k$ events occurring over a time interval of duration $t$ is a function that depends only on $k$ and $t$.

The property of "lack of aftereffect" means that the probability of $k$ events occurring in any time interval does not depend on whether events occurred or did not occur at moments of time preceding the start of the considered interval. In other words, the history of the flow does not affect the probabilities of events occurring in the near future.

The property of ordinarity means that the occurrence of two or more events over a small time interval is practically impossible. In other words, the probability of more than one event occurring over a small time interval is negligible compared to the probability of only one event occurring.

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

If the constant intensity of the flow $\lambda$ is known, then the probability of $k$ events occurring in the simplest flow over time $t$ is determined by the Poisson formula

$$
P_{t}(k)=\frac{(\lambda t)^{k} \cdot \mathrm{e}^{-\lambda t}}{k!}
$$

Remark. A flow that has the property of stationarity is called stationary; otherwise, it is called non-stationary.