Example 8. Two dice are rolled. What is the probability that the sum of the points on both dice will not exceed 5?
Solution. Let $n_{1}$ points fall on the first die, $n_{2}$ - on the second. The space of elementary events is the set of pairs $(n_{1}, n_{2})$:

$$
\Omega=\left\{\left(n_{1}, n_{2}\right): n_{1}, n_{2}=1,2,3,4,5,6\right\}
$$

Event $A$ is of the form

$$
A=\left\{\left(n_{1}, n_{2}\right): n_{1}, n_{2}=1,2,3,4 ; n_{1}+n_{2} \leq 5\right\} .
$$

The set $\Omega$ contains 36 elements (see Table 1.1), the set $A-10$ elements $((1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2)$, $(4,1))$. According to formula (1.7.5) we get

$$
P(A)=\frac{10}{36}=\frac{5}{18}
$$

## Problems

1. Two dice are tossed. Find the probability that the sum of the points on both dice is greater than 6 (event $A$)?
2. In a lottery, 100 tickets are raffled. The prize falls on 10 tickets. Someone bought 3 tickets. What is the probability that at least one of them will win?
3. In a box, there are 6 blue and 9 red balls. Three balls are drawn from the box. Find the probability that two of them will be blue.
4. On eight identical cards, the numbers 2, $4,6,7,8,11,12$ and 13 are written. Two cards are drawn at random. Determine the probability that the fraction formed from the two obtained numbers is reducible.
5. Out of ten tickets, two are winning. Find the probability that among the five tickets taken at random: a) one is winning; b) both are winning.

## Answers

$$
\text { 1. } \frac{21}{36}=\frac{7}{12} \cdot 2 \cdot \frac{C_{100}^{3}-C_{90}^{3}}{C_{100}^{3}}=1-\frac{C_{90}^{3}}{C_{100}^{3}}=1-\frac{90 \cdot 89 \cdot 88}{100 \cdot 99 \cdot 98} \approx 0.2735 \cdot 3 \cdot \frac{27}{91} \cdot 4 \cdot \frac{5}{14} \text {. }
$$

5. a) $\frac{5}{9}$; b) $\frac{2}{9}$.

## Questions

1. What is called the space of elementary events?
2. What is called an elementary event?
3. What is called an event?
4. What is called an impossible event?
5. What is called a certain event?

6. What is called a random event?

7. How are opposite events defined?
8. What are called incompatible events?
9. What is called a complete group of events?
10. What is called an algebra of events?
11. How is the probability of an event defined?
12. What are the axioms of probability?
13. What is called a probability space?
14. What is the sum of the probabilities of opposite events?

## § 1.8. Addition and Multiplication of Probabilities

## Theorem of Addition of Probabilities of Two Events

The probability of the sum of two events is equal to the sum of the probabilities of these events minus the probability of their joint occurrence:

$$
P(A+B)=P(A)+P(B)-P(A B)
$$

Theorem of Addition of Probabilities of Two Incompatible Events

The probability of the sum of two incompatible events is equal to the sum of the probabilities of these events:

$$
P(A+B)=P(A)+P(B)
$$

Remark 1. Formula (1.8.2) is derived from formula (1.8.1) when $A$ and $B$ are incompatible events; in this case $A B$ is an impossible event and $P(A B)=0$.

Theorem of Addition of Probabilities of $n$ Incompatible Events

The probability of the sum of $n$ incompatible events $A_{1}, A_{2}, \ldots, A_{n}$ is equal to the sum of the probabilities of these events:

$$
P\left(A_{1}+A_{2}+\ldots+A_{n}\right)=P\left(A_{1}\right)+P\left(A_{2}\right)+\ldots+P\left(A_{n}\right)
$$

The sum of the probabilities of events $A_{1}, A_{2}, \ldots, A_{n}$, forming a complete group, is equal to one:

$$
P\left(A_{1}\right)+P\left(A_{2}\right)+\ldots+P\left(A_{n}\right)=1
$$

The sum of the probabilities of opposite events is equal to one:

$$
P(A)+P(\bar{A})=1
$$

If we denote

$$
P(A)=p, P(\bar{A})=q
$$

then formula (1.8.5) takes the form

$$
p+q=1
$$

The probability of event $B$ given that event $A$ has occurred is called the conditional probability of event $B$ and is denoted as: $P(B / A)$, or $P_{A}(B)$.

## Theorem of Multiplication of Probabilities

The probability of the product of two events is equal to the product of the probability of one of them and the conditional probability of the other given that the first event has occurred:

$$
P(A B)=P(A) P(B / A), P(A B)=P(B) P(A / B)
$$

Event $B$ is independent of event $A$ if

$$
P(B / A)=P(B)
$$

i.e., the probability of event $B$ does not depend on whether event $A$ has occurred.

In this case, event $A$ is also independent of event $B$, i.e., the property of independence of events is mutual.

Note that if $A$ and $B$ are independent, then $\bar{A}$ and $B, A$ and $\bar{B}, \bar{A}$ and $\bar{B}$ are also independent.

Theorem of Multiplication of Probabilities of Two Independent Events

The probability of the product of two independent events is equal to the product of their probabilities:

$$
P(A B)=P(A) P(B)
$$

Theorem of Multiplication of Probabilities of $n$ Events

The probability of the product of $n$ events is equal to the product of one of them and the conditional probabilities of all the others, calculated under the assumption that all previous events have occurred:

$$
P\left(A_{1} A_{2} \ldots A_{n}\right)=P\left(A_{1}\right) P\left(A_{2} / A_{1}\right) P\left(A_{3} / A_{1} A_{2}\right) \ldots P\left(A_{n} / A_{1} A_{2} \ldots A_{n-1}\right)
$$

In particular, for three events $A, B, C$ formula (1.8.11) takes the form

$$
P(A B C)=P(A) P(B / A) P(C / A B)
$$

Events $A_{1}, A_{2}, \ldots, A_{n}$ are called independent in the aggregate, or independent, if they are pairwise independent, and also each of them is independent of the product of $k$ of the others ($k=2,3, \ldots, n-1$).

Remark 2. Pairwise independence of events does not imply their independence in the aggregate.

Remark 3. If events $A_{1}, A_{2}, \ldots, A_{n}$ are independent, then the opposite events $\bar{A}_{1}, \bar{A}_{2}, \ldots, \bar{A}_{n}$ are also independent.

## Theorem of Multiplication of Probabilities of $n$ Independent Events

If events $A_{1}, A_{2}, \ldots, A_{n}$ are independent, then the probability of their product is equal to the product of the probabilities of these events:

$$
P\left(A_{1} A_{2} \ldots A_{n}\right)=P\left(A_{1}\right) P\left(A_{2}\right) \ldots P\left(A_{n}\right)
$$

Remark 4. Equality (1.8.13) expresses the necessary and sufficient condition for the independence of events $A_{1}, A_{2}, \ldots, A_{n}$.

For three independent events $A, B, C$ formula (1.8.13) takes the form

$$
P(A B C)=P(A) P(B) P(C)
$$

The calculation of the probability of the sum of events can be reduced to the calculation of the probability of the product of opposite events by the formula

$$
P\left(A_{1}