Problem 3. In an urn, there are $a$ white and $b$ black balls. What is the probability that one of the two balls drawn is white and the other is black? (The drawn ball is not returned to the urn).
Solution. Let's denote the events:

$A$ - a white ball appears during the first draw;

$B$ - a black ball appears during the second draw;

$C$ - a black ball appears during the first draw;

## $D$ - a white ball appears during the second draw.

Then:

$A B$ - the first ball drawn is white, and the second is black;

$C D$ - the first ball drawn is black, and the second is white,

$A B+C D$ - the balls drawn are of different colors.

By formulas (6) and (9) we have:

$$
\begin{aligned}
p(A B+C D)= & p(A B)+p(C D)=p(A) p(B / A)+p(C) p(D / C)= \\
& =\frac{a}{a+b} \cdot \frac{b}{a+b-1}+\frac{b}{a+b} \cdot \frac{a}{a+b-1}=\frac{2 a b}{(a+b)(a+b-1)}
\end{aligned}
$$

Problem 4. Among 100 lottery tickets, there are 5 winning ones. Find the probability that two randomly purchased tickets are winning.

Solution. Let's denote the events $A$ - the first ticket is a winner; $B$ - the second ticket is a winner; $A B$ - both tickets are winners. Event $B$ depends on event $A$, so by formula (9) we have

$$
p(A B)=p(A) p(B / A)=\frac{5}{100} \cdot \frac{4}{99}=\frac{1}{495}
$$

Problem 5. Three shooters shoot at a target independently of each other. The probability of hitting the target for the first shooter is 0.6; for the second, 0.7; for the third, 0.75. Find the probability of at least one hit in the target, if each shooter makes one shot.

Solution. Introduce events $A, B$, and $C$, which consist in the first, second, and third shooters hitting the target, respectively. Event $A+B+C$ means: "at least one shooter hits the target" (see $\$ 2$, point $3^{\circ}$). Events $A, B, C$ are independent in combination, so by formula (12) we get:

$$
\begin{gathered}
p(A+B+C)=1-(1-0.6)(1-0.7)(1-0.75)= \\
\quad=1-0.4 \cdot 0.3 \cdot 0.25=0.97
\end{gathered}
$$