Solution Found!
For the k-out-of-n system described in Problem, assume
Chapter 3, Problem 17STE(choose chapter or problem)
For the k-out-of-n system described in Problem 3.67, assume that each component independently works with probability \(\frac{1}{2}\). Find the conditional probability that component 1 is working, given that the system works, when
(a) k = 1, n = 2;
(b) k = 2, n = 3.
Problem 3.67
An engineering system consisting of n components is said to be a k-out-of-n system (k … 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 ith 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.
Questions & Answers
QUESTION:
For the k-out-of-n system described in Problem 3.67, assume that each component independently works with probability \(\frac{1}{2}\). Find the conditional probability that component 1 is working, given that the system works, when
(a) k = 1, n = 2;
(b) k = 2, n = 3.
Problem 3.67
An engineering system consisting of n components is said to be a k-out-of-n system (k … 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 ith 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.
ANSWER:Step 1 of 2
(a)
Assume that each component independently works with probability
We are asked to find the conditional probability that component 1 is working, given that the system works, when
Let be the event the
component is working and for each part let
be the event at least
of
components are working.
To have one of two components working means that
Thus we have using addition rule of probability,
Let
Then the probability of the event of interest or can be computed as
[We have given each component independently works with probability ]
Since if is true then
must also be true, hence
Hence the conditional probability that component 1 is working, given that the system works, when is