An engineering system consisting of n components is said to be a k-out-of-n system (k≤n) when the system functions if and only if at least k out of the n components function. Suppose that all components function independently of each other and are not identical. If the ith component functions with probability pi, i=1,2,3,4, compute the probability that a 2-out-of-4 system functions.
Option 1: $$ \Bbb ℙ(2,3,4) $$
$$=P_1 [1−(1−P_2)(1−P_3)(1−P_4)]+ $$ $$P_2 [1−(1−P_1)(1−P_3)(1−P_4)]+ $$ $$P_3 [1−(1−P_1)(1−P_2)(1−P_4)]+ $$ $$P_4 [1−(1−P_1)(1−P_2)(1−P_3)] $$
Would it be the same as 4-choose-2? $$ \Bbb ℙ(2,3,4) $$ $$ \Bbb =P_1 P_2 P_3 P_4 $$ $$ - P_1P_2 - P_1P_3 - P_1P_4 - P_2P_3 - P_2P_4 - P_3P_4 - P_1P_2P_3P_4 $$