Solution. Let $A, B, C, D$ denote the events of drawing the first, second, third, and fourth letter of the word "papa" from a set of 6 letters: $a, a, a, n, n, x$. We need to find the probabilities of the events: $A, B / A, C / A B, D / A B C$.

$$
P(A)=\frac{2}{6}=\frac{1}{3}
$$

$$
\begin{aligned}
& P(B / A)=\frac{3}{5} \\
& P(C / A B)=\frac{1}{4} \\
& P(D / A B C)=\frac{2}{3}
\end{aligned}
$$

According to formula (1.8.11) for $n=4$, we get

$$
P(A B C D)=P(A) P(B / A) P(C / A B) P(D / A B C)=\frac{1}{3} \cdot \frac{3}{5} \cdot \frac{1}{4} \cdot \frac{2}{3}=\frac{1}{30}
$$

## Problems

1. A company produces on average $25 \%$ of top-grade products and $65 \%$ of first-grade products. What is the probability that a randomly selected item will be of first or top grade?
2. Each of four mutually exclusive events can occur with probabilities $0.014, 0.011, 0.009, 0.006$. Find the probability that at least one of these events will occur as a result of the experiment.
3. A shooter fires one shot at a target consisting of a central circle and two concentric rings. The probabilities of hitting the circle and the rings are $0.35, 0.20, 0.15$ respectively. What is the probability of hitting the target?
4. Determine the probability that a randomly selected two-digit number will be divisible by either 2, 9, or both.
5. Find the probability that when a die is rolled, an even or a multiple of three number of points will appear on the top face.
6. On ten cards, the digits from 0 to 9 are printed. Determine the probability that three randomly selected and placed in a row cards will form the number 357.
7. Four shots are fired at a target with a probability of hitting 0.2 for each shot. Hits on the target at different shots are assumed to be independent events. What is the probability of hitting the target exactly three times?
8. The probabilities of the occurrence of each of three independent events $A_{1}, A_{2}, A_{3}$ are $p_{1}=0.9, p_{2}=0.8, p_{3}=0.7$ respectively. Find the probability of the occurrence of only one of these events:
9. All faces of a die are covered with opaque paper: faces $1,2,3$ - blue, faces 4, 5,6 - red. When the die is rolled, a red face appears. Determine the probability that this face has an even number.
10. The word "lotus," made of letter-cubes, is scattered into individual letters, which are then mixed and placed in a box. Find the probability that when three letters are randomly drawn one after another, the word "sto" will appear.
11. In a box, there are 10 parts, 4 of the first type and 6 of the second. To assemble an aggregate, a part of the first type must be taken first, and then a part of the second type. What is the probability that the parts will be taken in the required sequence when selected at random.
12. The probability that an event will occur at least once in three independent trials is 0.875. Find the probability of the event occurring in one trial.
13. A student knows the answers to 20 questions out of 26. Suppose the questions are asked one after another. Find the probability that three consecutive questions will be lucky.
14. In a box, there are 10 parts, 5 of the first type, 3 of the second, and 2 of the third. What is the probability that when selected at random, the first will be a part of the first type, the second of the second type, and the third of the third type.
15. Find the probability that a randomly selected item will be first-grade, given that $4 \%$ of the total production is defective, and $75 \%$ of the non-defective items meet the requirements of the first grade.
16. A batch of one hundred parts is subject to selective inspection. The condition for the rejection of the entire batch is the presence of at least one defective part among the five inspected. What is the probability that the batch will be rejected if it contains $5 \%$ defective parts.
17. A coin and a die are tossed. Find the probability that the coin will show a number, and the die will show a number of points that is a multiple of three.
18. The probability that a shooter will hit the bullseye with one shot is 0.6. How many shots must the shooter take to ensure a probability of at least 0.8 that he will hit the bullseye at least once?

## Answers

1. 0.9. 2. 0.4. 3. 0.7. 4. 5/9. 5. 2/3. 6. 1/720. 7. 0.0256. Indication. The event "hitting the target exactly three times out of four shots" is favorable in 4 outcomes: $A A A \bar{A}, A A \bar{A} A, A \bar{A} A A, \bar{A} A A A$, where $A$ is a hit, $\bar{A}$ is a miss (the notation $A A A \bar{A}$ means a hit on the first, second, and third shots, and a miss on the fourth shot). The probability of each outcome is the same: $0.2 \cdot 0.2 \cdot 0.2 \cdot 0.8=0.064$, where $0.8=1-0.2$ is the probability of a miss. The desired probability $p=0.0256=4 \cdot 0.064$. 8. 0.092. Indication. If we denote $B_{1}=A_{1} \bar{A}_{2} \bar{A}_{3}, B_{2}=\bar{A}_{1} A_{2} \bar{A}_{3}, B_{3}=\bar{A}_{1} \bar{A}_{2} A_{3}$; then the desired probability $p=P\left(B_{1}+B_{2}+B_{3}\right)=P\left(B_{1}\right)+P\left(B_{2}\right)+P\left(B_{3}\right) .9 .2 / 3$. Indication. Find the conditional probability $P(A / B)$, where event $A$ is "an even number of points appears," and event $B$ is "a number of points greater than 3 appears." 10. 1/30. Indication. Find the probability of the event $A=A_{1} A_{2} A_{3}$, where $A_{1}$ is "the first letter drawn is $c$", $A_{2}$ is "the second letter drawn is $m$", $A_{3}$ is "the third letter drawn is $o$". 11. $4 / 15.12 .0.5$. Indication. Apply formula (1.8.17). 13. 57/115. Indication. Apply formula (1.8.12). 14. 1/24. 15. 0.72. 16. 0.23. Indication. First find the probability $q$ of the opposite event $A$, which consists in the acceptance of the batch of parts. This event is the product of five events $A=A_{1} A_{2} A_{3} A_{4} A_{5}$, where $A_{k}(k=1,2,3,4,5)$ means that the $k$-th inspected part is standard. Further, $P\left(A_{1}\right)=95 / 100$, $P\left(A_{2} / A_{1}\right)=94 / 99$ and so on. 17. 1/6. 18. $n \geq 2$.

## Questions

1. What is the probability of the sum of two events?
2. What is the probability of the sum of two mutually exclusive events?
3. State the theorem on the probability of the sum of $n$ mutually exclusive events.
4. What is the sum of the probabilities of events forming a complete group?
5. What is the sum of the probabilities of opposite events?
6. State the theorem on the probability of the product of two events.
7. How is the independence of two events defined?
8. What is the probability of the product of two independent events?
9. State the theorem on the probability of the product of $n$ events?
10. How is the independence of $n$ events defined?
11. What is the probability of the product of $n$ independent events?
12. How to
Example 25. The word papakha is made up of letters from a cut-out alphabet. The cards with the letters are thoroughly shuffled. Four cards are drawn in sequence and laid out in a row. What is the probability of getting the word nana this way?