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.

Question

Assume that each component independently works with probability 1/2. Find the conditional probability that component 1 is working, given that the system works, when k = 2, n = 3.

Answer 1

p = probability component works = 1/2

q = probability component doesn't work = 1 - 1/2 = 1/2

If the system works when k = 2 and n = 3, that means that 2 out of the 3 components need to work. The possibilities are as follows:

1) component 1 works, component 2 works, component 3 doesn't work: P = p * p * q = (1/2)(1/2)(1/2) = 1/8

2) component 1 works, component 2 doesn't work, component 3 works: P = p * q * p = (1/2)(1/2)(1/2) = 1/8

3) component 1 doesn't work, component 2 works, component 3 works: P = q * p * p = (1/2)(1/2)(1/2) = 1/8

The probability that component 1 is working given that the system works would be the probability that situation 1 and situation 2 happen. To find the desired conditional probability, we would add the probability of situation 1 happening to the probability of situation 2 happening:

P(component 1 works given system works) = (1/8) + (1/8) = 2/8 = 1/4

We can use a binomial probability to check:

P = (_nC_k) p^k q^n-k = (₃C₂) (1/2)² (1/2)^3-2 = 3 (1/2)² (1/2)¹ = 3 (1/4) (1/2) = 3/8

This would be the probability that the system works. Since component 1 works in only 2 out of 3 conditions where the system works, we can multiply our binomial probability by 2/3 to get the desired conditional probability:

P(component 1 works given system works) = (2/3) * (3/8) = 2/8 = 1/4