Example 7. From the first machine, 200 parts were sent to assembly, of which 190 are standard; from the second - 300, of which 280 are standard. Find the probability of event $A$, which consists in a randomly taken part being standard, and the conditional probabilities of it relative to events $B$ and $\bar{B}$, if event $B$ consists in the part being manufactured on the first machine.
Solution. The probability of event $A$ is the ratio of the number of all standard parts to the total number of parts manufactured on both machines

$$
P(A)=\frac{190+280}{200+300}=\frac{470}{500}=0.94
$$

The conditional probability of event $A$ given event $B$ (the probability that a randomly taken part is standard, given that it was manufactured on the first machine)

$$
P(A / B)=190 / 200=0.95
$$

The conditional probability of event $A$ given event $\bar{B}$, i.e., the probability that a taken part is standard, given that it was not manufactured on the first (but on the second) machine

$$
P(A / \bar{B})=\frac{280}{300}=\frac{14}{15} \approx 0.93
$$