724. The system consists of two blocks connected in series. The system fails if at least one block fails. The first block contains two elements: $\boldsymbol{A}, \boldsymbol{B}$ (they are connected in parallel) and fails when both elements fail simultaneously. The second block contains one element $C$ and fails when this element fails. a) Find an estimate $P^*$ of the reliability (probability of failure-free operation) of the system using the Monte Carlo method, given the probabilities of failure-free operation of the elements: $P(A)=0.8$,
$P(B)=0.85, P(C)=0.6 ;$ b) find the absolute error $\left|P-P^{*}\right|$, where $P$ is the reliability of the system, calculated analytically. Conduct 50 trials.
The solution is as follows. a) We will select three random numbers from Table 9: 0.10, 0.09, and 0.73; according to the rule (if the random number is less than the probability of the event, the event occurs; if the random number is greater than or equal to the probability of the event, the event does not occur), we will simulate events \(A\), \(B\), and \(C\), which consist of the failure-free operation of elements \(A\), \(B\), and \(C\) respectively. The results of the test will be recorded in Calculation Table 57.

Since \(P(A) = 0.8\) and \(0.10 < 0.8\), event \(A\) occurs, i.e., element \(A\) works without failure. Similarly, since \(P(B) = 0.85\) and \(0.09 < 0.85\), event \(B\) occurs, i.e., element \(B\) works without failure. Since \(P(C) = 0.6\) and \(0.73 > 0.6\), event \(C\) does not occur, i.e., element \(C\) fails; in other words, the second block, and thus the entire system, fails. We place a minus sign in the corresponding cells of Table 57.

Similarly, the other tests are simulated. Table 57 presents the results of four tests.

|  |  |  |  |  |  |  |  |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  |  |  |  | Elements |  |  | Blocks | System |
|  |  | \(\bar{A}\) | \(\bar{B}\) | \(\bar{C}\) | A | B | \(\bar{C}\) |  |  |
| 1 | First  Second | 0.10 | 0.09 | 0.73 | + | + | - | \(\pm\) | - |
| 2 | First  Second | 0.25 | 0.33 | 0.76 | + | + | - | \(\pm\) | - |
| 3 | First  Second | 0.52 | 0.01 | 0.35 | + | + | + | + | + |
| 4 | First  Second | 0.86 | 0.34 | 0.67 | - | + | - | \(\pm\) |  |

By conducting 50 tests, we find that the system worked without failure in 28 of them. As an estimate of the desired reliability \(P\), we take the relative frequency \(P^{*} = 28 / 50 = 0.56\).

b) We will find the reliability of the system \(P\) analytically. The probabilities of failure-free operation of the first and second blocks are respectively:

\[
P_{1} = 1 - P(\bar{A}) \cdot P(\bar{B}) = 1 - 0.2 \cdot 0.15 = 0.97, \quad P_{2} = P(C) = 0.6
\]

The probability of failure-free operation of the system is

\[
P = P_{1} \cdot P_{2} = 0.97 \cdot 0.6 = 0.582
\]

The desired absolute error \(\left|P - P^{*}\right| = 0.582 - 0.56 = 0.022\).