Example 9. A radio tube may belong to one of three batches with probabilities: $p_{1}=0.2, p_{2}=0.3, p_{3}=0.5$. The probability that the tube will work for a given number of hours, for these batches, is respectively: 0.9; 0.8; 0.7. Determine the probability that the radio tube will work for the given number of hours.
Solution. Let's introduce the following notations: $A$ - the event that the lamp will work for the specified number of hours; $H_{1}, H_{2}, H_{3}$ - hypotheses that the lamp belongs to the first, second, and third batch, respectively.

According to the problem, we have:

$$
\begin{aligned}
& P\left(H_{1}\right)=0.2, \quad P\left(H_{2}\right)=0.3, \quad P\left(H_{3}\right)=0.5 \\
& P\left(A / H_{1}\right)=0.9, \quad P\left(A / H_{2}\right)=0.8, \quad P\left(A / H_{3}\right)=0.7
\end{aligned}
$$

The formula for the total probability (1.9.1) for $n=3$ is given by

$$
P(A)=P\left(H_{1}\right) P\left(A / H_{1}\right)+P\left(H_{2}\right) P\left(A / H_{2}\right)+P\left(H_{3}\right) P\left(A / H_{3}\right)
$$

According to this formula, we find the desired probability

$$
P(A)=0.2 \cdot 0.9+0.3 \cdot 0.8+0.5 \cdot 0.7=0.77
$$