Q. 3.67

Question

An engineering system consisting of $n$ components is said to be a $k$ -out-of- $n$ system $(k \leq n)$ if the system functions if and only if at least $k$ of the $n$ components function. Suppose that all components function independently of one another. (a) If the component functions with probability $p_{i}, i = 1, 2, 3, 4,$ , compute the probability that a $2$ -out-of- $4$ system functions. (b) Repeat part (a) for a $3$ -out-of- $5$ system. (c) Repeat for a k-out-of-n system when all the Pi equal p (that is, $p_{i}, i = 1, 2, . . . . . n)$

Step-by-Step Solution

Verified

Answer

a) The probability of $2 -$ out of $- 4$ system function is

$1 - Q_{1} Q_{2} Q_{3} Q_{4} - P_{1} Q_{2} Q_{3} Q_{4} - Q_{1} P_{2} Q_{3} Q_{4} - Q_{1} Q_{2} P_{3} Q_{4} - Q_{1} Q_{2} Q_{3} P_{4}$

b) The Probability of $3 -$ out of5 system function $A + B + C$ where

$\begin{aligned} A = Q_{1} Q_{2} P_{3} P_{4} P_{5} + Q_{1} P_{2} Q_{3} P_{4} P_{5} + Q_{1} P_{2} P_{3} Q_{4} P_{5} + Q_{1} P_{2} P_{3} P_{4} Q_{5} + P_{1} Q_{2} Q_{3} P_{4} P_{5} + \\ + P_{1} Q_{2} P_{3} Q_{4} P_{5} + P_{1} Q_{2} P_{3} P_{4} Q_{5} + P_{1} P_{2} Q_{3} Q_{4} P_{5} + P_{1} P_{2} Q_{3} P_{4} Q_{5} + P_{1} P_{2} P_{3} Q_{4} Q_{5} \end{aligned}$

$B = Q_{1} P_{2} P_{3} P_{4} P_{5} + P_{1} Q_{2} P_{3} P_{4} P_{5} + P_{1} P_{2} Q_{3} P_{4} P_{5} + P_{1} P_{2} P_{3} Q_{4} P_{5} + P_{1} P_{2} P_{3} P_{4} Q_{5}$

$C = P_{1} P_{2} P_{3} P_{4} P_{5}$

1Step 1: Component Function

a) The events name was given below,

$S = \{t h e 2 - o u t o f - 4 s y s t e m f u n c t i o n s\}$

$A_{i} = {the i -th component functions}, i = 1, 2, 3, 4 .$

$B_{k} = {exactly k components function}, k = 0, 1, 2, 3, 4,$

$S = {at least 2 components functions} = B_{2} \cup B_{3} \cup B_{4} .$

The disjoint sets are $i \neq j,$ we have $S^{c} = B_{0} \cup B_{1}$

2Step 2: System Function

we mentioned the equations, $Q_{i} = 1 - P_{i} = P (A_{i}^{c})$

$P (B_{0}) = P (A_{1}^{c} \cap A_{2}^{c} \cap A_{3}^{c} \cap A_{4}^{c}) = [independence of A_{i}]$

$\begin{aligned} = P (A_{1}^{c}) P (A_{2}^{c}) P (A_{3}^{c}) P (A_{4}^{c}) \end{aligned}$

$= Q_{1} Q_{2} Q_{3} Q_{4} .$

$P (B_{1}) = P (\cup_{i = 1}^{4} \underset{j \neq i}{\cap} A_{i} A_{j}^{c}) = [aditivity]$

$= \sum_{i = 1}^{4} P (\underset{j \neq i}{\cap} A_{i} A_{j}^{c}) = [independence of A_{i}]$

$= \sum_{i = 1}^{4} P (A_{i}) \prod_{j \neq i} P (A_{j}^{c}) = \sum_{i = 1}^{4} P_{i} \prod_{j \neq i} Q_{j}$

$= P_{1} Q_{2} Q_{3} Q_{4} + Q_{1} P_{2} Q_{3} Q_{4} + Q_{1} Q_{2} P_{3} Q_{4} + Q_{1} Q_{2} Q_{3} P_{4}$

The chances of $2 - o u t o f - 4$ System function is

$P (S) = 1 - P (S^{c})$

$= 1 - P (B_{0} \cup B_{1})$

$= 1 - [P (B_{0}) + P (B_{1})]$

$= 1 - Q_{1} Q_{2} Q_{3} Q_{4} - P_{1} Q_{2} Q_{3} Q_{4} - Q_{1} P_{2} Q_{3} Q_{4} - Q_{1} Q_{2} P_{3} Q_{4} - Q_{1} Q_{2} Q_{3} P_{4}$

3Step 3: Independent Event

we mentioned the event name,

$S = \{t h e 3 - o u t o f - 5 s y s t e m f u n c t i o n s\}$

$A_{i} = {the i -th component functions}, i = 1, 2, 3, 4 . 5$

$B_{k} = {exactly k components function}, k = 0, 1, 2, 3, 4, 5$

$S = {at least 3 components functions} = B_{3} \cup B_{4} \cup B_{5} .$

Twice the disjoint sets are, $B_{k}$

4Step 4: probability of system function

we mentioned $Q_{i} = 1 - P_{i} = P (A_{i}^{c})$

$P (B_{3}) = P (\underset{1 \leq i, j, k \leq 5}{\cup} \underset{m, n \neq i, j, k}{\cap} A_{i} A_{j} A_{k} A_{m}^{c} A_{n}^{c})$

$P (B_{4}) = P (\cup_{i = 1}^{5} \underset{j \neq i}{\cap} A_{i}^{c} A_{j})$

$\begin{aligned} = Q_{1} P_{2} P_{3} P_{4} P_{5} + P_{1} Q_{2} P_{3} P_{4} P_{5} + P_{1} P_{2} Q_{3} P_{4} P_{5} + \\ + P_{1} P_{2} P_{3} Q_{4} P_{5} + P_{1} P_{2} P_{3} P_{4} Q_{5} \end{aligned}$

$P (B_{5}) = P (A_{1} A_{2} A_{3} A_{4} A_{5}) = P_{1} P_{2} P_{3} P_{4} P_{5}$

The probability that a $3 - o u t o f - 5$ system function is,

$P (S) = P (B_{3} \cup B_{4} \cup B_{5})$

$= P (B_{5}) + P (B_{4}) + P (B_{5})$

$\begin{aligned} = Q_{1} Q_{2} P_{3} P_{4} P_{5} + Q_{1} P_{2} Q_{3} P_{4} P_{5} + Q_{1} P_{2} P_{3} Q_{4} P_{5} + Q_{1} P_{2} P_{3} P_{4} Q_{5} + \\ + P_{1} Q_{2} Q_{3} P_{4} P_{5} + P_{1} Q_{2} P_{3} Q_{4} P_{5} + P_{1} Q_{2} P_{3} P_{4} Q_{5} + P_{1} P_{2} Q_{3} Q_{4} P_{5} + \\ + P_{1} P_{2} Q_{3} P_{4} Q_{5} + P_{1} P_{2} P_{3} Q_{4} Q_{5} + \\ + Q_{1} P_{2} P_{3} P_{4} P_{5} + P_{1} Q_{2} P_{3} P_{4} P_{5} + P_{1} P_{2} Q_{3} P_{4} P_{5} + \\ + P_{1} P_{2} P_{3} Q_{4} P_{5} + P_{1} P_{2} P_{3} P_{4} Q_{5} + \\ + P_{1} P_{2} P_{3} P_{4} P_{5} . \end{aligned}$

Q. 3.66

Q.3.75

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