Example. The probability of failure of the $i$-th conducting element of the circuit within time $T$, as shown in Fig. 6.3, is $p_{i}$. All elements of the circuit function independently of each other. Find the probability that the entire circuit will not fail within time $T$, i.e., that current can flow through the circuit.

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-267.jpg?height=153&width=384&top_left_y=807&top_left_x=457)

Fig. 6.3
Solution. Let us introduce random events

$$
\begin{aligned}
& A=\left\{\begin{array}{c}
\text { the entire chain will not fail } \\
\text { during time } T
\end{array}\right\}, \\
& A_{i}=\left\{\begin{array}{c}
i \text {-th element of the chain will not fail } \\
\text { during time } T
\end{array}\right\}, i=1,2,3,4 .
\end{aligned}
$$

By the condition, events $A_{1}, A_{2}, A_{3}, A_{4}$ are independent and $\mathrm{P}\left(A_{i}\right)=1-p_{i}$, $i=1,2,3,4$. Let's express event $A$ in terms of $A_{1}, A_{2}, A_{3}, A_{4}$. From Fig. 6.3, it is clear that

$$
A=A_{1} \cap\left(A_{2} \cup A_{3}\right) \cap A_{4} .
$$

Events $A_{1}, A_{2} \cup A_{3}, A_{4}$ are independent, so

$$
\mathrm{P}(A)=\mathrm{P}\left(A_{1}\right) \mathrm{P}\left(A_{2} \cup A_{3}\right) \mathrm{P}\left(A_{4}\right) .
$$

Events $A_{2}$ and $A_{4}$ are independent, so

$$
\mathrm{P}\left(A_{2} \cup A_{3}\right)=1-\left(1-\mathrm{P}\left(A_{2}\right)\right)\left(1-\mathrm{P}\left(A_{3}\right)\right) \text {. }
$$

From the last two formulas, we find that

$$
\mathrm{P}(A)=\mathrm{P}\left(A_{1}\right)\left[1-\left(1-\mathrm{P}\left(A_{2}\right)\right)\left(1-\mathrm{P}\left(A_{3}\right)\right)\right] \mathrm{P}\left(A_{4}\right)
$$

From this, considering that $\mathrm{P}\left(A_{i}\right)=1-p_{i}$, we obtain the desired probability:

$$
\mathrm{P}(A)=\left(1-p_{1}\right)\left(1-p_{2} p_{3}\right)\left(1-p_{4}\right)
$$

Answer. $\left(1-p_{1}\right)\left(1-p_{2} p_{3}\right)\left(1-p_{4}\right)$.

Conditions of the Problem. The probability of failure of the $i$-th element of the specified chain during some time $T$ is $p_{i}$. All elements of the chain function independently of each other. Find the probability that the entire chain will not fail during time $T$.

1. 

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-268.jpg?height=157&width=292&top_left_y=740&top_left_x=259)

3. 

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-268.jpg?height=108&width=293&top_left_y=925&top_left_x=258)

5. 

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-268.jpg?height=109&width=329&top_left_y=1083&top_left_x=240)

7. 

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-268.jpg?height=153&width=290&top_left_y=1280&top_left_x=260)

9. 

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-268.jpg?height=50&width=429&top_left_y=1540&top_left_x=194)

2. 

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-268.jpg?height=155&width=206&top_left_y=743&top_left_x=792)

4. 

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-268.jpg?height=54&width=336&top_left_y=950&top_left_x=733)

6. 

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-268.jpg?height=150&width=284&top_left_y=1066&top_left_x=759)

8. 

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-268.jpg?height=200&width=208&top_left_y=1255&top_left_x=793)

10. 

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-268.jpg?height=151&width=338&top_left_y=1489&top_left_x=732)

Answers. $\quad 1 .\left(1-p_{1} p_{2}\right)\left(1-p_{3}\right) . \quad 2.1-p_{1} p_{2} p_{3} . \quad 3.1-[1-$ $\left.-\left(1-p_{1}\right)\left(1-p_{2}\right)\right] p_{3}$. 4. $\left(1-p_{1}\right)\left(1-p_{2}\right)\left(1-p_{3}\right)$. 5. $\left(1-p_{1} p_{2}\right)\left(1-p_{3} p_{4}\right)$. 6. $\left(1-p_{1} p_{2} p_{3}\right)\left(1-p_{4}\right) .7 .1-p_{1} p_{2}\left[1-\left(1-p_{3}\right)\left(1-p_{4}\right)\right] . \quad 8.1-p_{1} p_{2} p_{3} p_{4}$. 9. $\left(1-p_{1}\right)\left(1-p_{2}\right)\left(1-p_{3}\right)\left(1-p_{4}\right) . \quad 10.1-p_{4}\left[1-\left(1-p_{1}\right)\left(1-p_{2} p_{3}\right)\right]$.

## 6.7. Bernoulli Scheme: Fixed Number of Trials

Problem Statement. Independent trials are conducted, in each of which there are two possible outcomes: success (S) and failure (F), with the probability of success $p$ being the same in all trials, and the probability of failure being $q(p+q=1)$. Find the probability of a certain random event related to the number of successes in each of the $n$ trials.

Remark. The sequence of trials described in the condition is called the Bernoulli scheme, and the trials themselves are sometimes called Bernoulli trials. Examples of the Bernoulli scheme include throwing a ball into a basketball hoop (success - hit, failure - miss), buying lottery tickets (success - winning ticket, failure - losing ticket), etc.

Plan of Solution. Each result of conducting $n$ trials can be represented by a "chain" of symbols S and F, for example, SSFFS...F. Due to the independence of the trials, the probability of each specific chain is $p^{k} q^{n-k}$, where $k$ is the number of symbols S in the chain.

To find the probability of the original event, one should select all chains leading to the given event and sum their probabilities.

The Bernoulli formula for the number of successes $\mu_{n}$ in $n$ Bernoulli trials is often used:

$$
\mathrm{P}\left(\mu_{n}=k\right)=C_{n}^{k} p^{k} q^{n-k}, \quad k=0,1, \ldots, n
$$