Example 8. At a distribution base, there are electric bulbs manufactured by two factories. Among them, $60 \%$ are made by the first factory and $40 \%$ by the second. It is known that out of every 100 bulbs made by the first factory, 95 meet the standard, and out of 100 bulbs made by the second factory, 85 meet the standard. Determine the probability that a randomly selected bulb will meet the standard.
Solution. Let $A$ be the event that the randomly selected bulb is standard. We introduce two hypotheses: $H_{1}$ "the bulb is manufactured by the first factory", $H_{2}$ - "the bulb is manufactured by the second factory".

From the conditions of the problem, it follows that

$$
\begin{aligned}
& P\left(H_{1}\right)=0.6, \quad P\left(H_{2}\right)=0.4 \\
& P\left(A / H_{1}\right)=0.95, \quad P\left(A / H_{2}\right)=0.85
\end{aligned}
$$

In accordance with formula (1.9.1) for $n=2$, we get

$$
P(A)=0.6 \cdot 0.95 + 0.4 \cdot 0.85 = 0.57 + 0.34 = 0.91
$$