Example 4. A sportsman shoots at a target divided into 3 sectors. The probability of hitting the first sector is 0.4, the second - 0.3. What is the probability of hitting either the first or the second sector?
Solution. Events $A$ - "hitting the first sector" and $B$ - "hitting the second sector" are mutually exclusive (hitting one sector excludes hitting the other), so the addition rule for probabilities of mutually exclusive events applies. According to this rule, the required probability is:

$$
P(A+B)=P(A)+P(B)=0.4+0.3=0.7
$$

Example 5. The probability of hitting the target for the first athlete is 0.85, and for the second athlete, it is 0.8. The athletes each made one shot independently of each other. Find the probability that at least one athlete will hit the target?

Solution. Let's introduce the notations: event $A$ - "the first athlete hits the target", $B$ - "the second athlete hits the target", $C$ - "at least one athlete hits the target". Clearly, $A+B=C$, and events $A$ and $B$ are compatible. According to formula (1.8.1), we get

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

or

54

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

since $A$ and $B$ are independent events, and formula (1.8.10) applies to them. Substituting the given values $P(A)=0.85, P(B)=0.8$ into the formula for $P(C)$, we find the required probability

$$
P(C)=(0.85+0.8)-0.85 \cdot 0.8=0.97
$$