Example. On average, 6 letters are sent to the department per week, of which 2 are from abroad. Find the probability that 13 letters will be sent in 2 weeks, of which 5 are from abroad.
SOLUTION. There are two independent simple streams of letters with intensities $\lambda_{1}=4$ (internal) and $\lambda_{2}=2$ (foreign). Let $\mu_{t}$ and $\tilde{\mu}_{t}$ be the number of events that occurred over time $t$ in the first and second streams, respectively. We are interested in the probability of the random event $\left\{\mu_{2}=8, \tilde{\mu}_{2}=5\right\}$. Using formula (1), we get

$$
\begin{aligned}
& \mathrm{P}\left(\mu_{2}=8\right)=\frac{8^{8}}{8!} e^{-8} \approx 0.1396 \\
& \mathrm{P}\left(\tilde{\mu}_{2}=5\right)=\frac{4^{5}}{5!} e^{-4} \approx 0.1563
\end{aligned}
$$

(approximate values of the probabilities can be obtained from Table 5.3 in the book by L.N. Bolshev and N.V. Smirnov "Tables of Mathematical Statistics", Moscow: Nauka, 1983 or using the PEШЕБНИK.ВM package). Since the random events $\left\{\mu_{2}=8\right\}$ and $\left\{\tilde{\mu}_{2}=5\right\}$ are independent, we find the desired probability from relations (2) and (3)

$$
\mathrm{P}\left(\mu_{2}=8, \tilde{\mu}_{2}=5\right)=\mathrm{P}\left(\mu_{2}=8\right) \mathrm{P}\left(\tilde{\mu}_{2}=5\right) \approx 0.1396 \cdot 0.1563 \approx 0.0218
$$

Answer. $\mathrm{P}\left(\mu_{2}=8, \tilde{\mu}_{2}=5\right) \approx 0.0218$.

PROBLEM CONDITIONS.

1. In a certain city, the average number of road traffic accidents per day is 5. Find the probability that there will be 10 road traffic accidents over two days.
2. In Rutherford and Geiger's observations, a radioactive substance emitted on average $3.87 \alpha$-particles over a period of 7.5 seconds. Find the probability that this substance will emit at least one $\alpha$-particle in one second.
3. Calls are made to an information bureau on average 30 times per hour. Find the probability that the first call after the start of the shift will occur no later than 1 minute.
4. Calls are made to an information bureau on average 30 times per hour. Find the probability that the second call after the start of the shift will occur no earlier than 1 minute and no later than 2 minutes.
5. Buses on a certain route arrive at a specific stop on average every 15 minutes. Find the probability that the waiting time for a bus will be at least 20 minutes.
6. A certain amount of dough is sprinkled with 1000 raisins, after which everything is thoroughly mixed. From this dough, 100 buns are made. Find the probability that a specific bun will contain exactly 10 raisins.
7. The average density of pathogenic bacteria in one cubic meter of air is 80. A sample of 1 cubic decimeter of air is taken. Find the probability that at least one bacterium will be found in it.
8. A book of 500 pages contains 500 typos. Find the probability that a specific page will contain at least three typos.
9. Messages to an office arrive either by phone or by fax. On average, 20 messages arrive per hour, with phone calls being three times more frequent than faxes. What is the probability that the number of phone calls in an hour will be more than 15, and the number of faxes will be less than 5?
10. During an eight-hour workday, on average, 10 orders for the production of a certain product are received from Moscow and 5 orders from St. Petersburg. Find the probability that at least one order from St. Petersburg and no orders from Moscow will be received during an hour of work.

Answers. 1. 0.1251. 2. 0.4031. 3. 0.3935. 4. 0.1740. 5. 0.2636. 6. 0.1251. 7. 0.0769. 8. 0.0803. 9. 0.1903. 10. 0.1331.

### 6.11. Formulas of Total Probability and Bayes

PROBLEM STATEMENT. Consider a situation where a second random experiment is conducted following a certain random experiment, based on the results of the first.

1. Find the probability of a certain random event $A$ associated with the second experiment.
2. It is known that event $A$ occurred. Find the probability that the first experiment ended with a certain outcome.

PLAN OF SOLUTION.