## 133. Math Puzzle $6 / 76$

A catch device that secures the hoist basket in a shaft fails in at most one out of 1000 operational cases. Another independent safety device fails in at most one out of 100 cases where it is called upon.

What is the probability that the occupants will be saved by the safety devices if the hoist system fails?
The probability that the two (independent) safety devices of the hoist basket fail simultaneously is, according to the and-rule:

$$
P_{\text {Failure }}=P_{1} \cdot P_{2}=\frac{1}{1000} \cdot \frac{1}{100}=10^{-5}=0.001 \%
$$

Thus, the probability that the safety system functions is

$$
P=1-P_{\text {Failure }}=1-10^{-5}=0.99999=99.999 \%
$$

This means that it can almost be considered certain $(P=1)$ that the safety system will function.