Independently Dependent

Probability Level 5

A system consists of a set S S of n n independent components. A component a i ∈ S a_i \in S is either working (with probability p i p_i ), or has failed (with probability 1 − p i 1-p_i ), ∀ i \forall i . In addition, there is a family of m m component subsets { C k ⊆ S } , k = 1 , 2 , … m , \{C_k \subseteq S\}, k=1,2,\ldots m, such that the overall system works, if and only if, at least one of the components in each of these subsets is working. Let w w denote the probability that the overall system is working. Which of the given statements is always true regardless the structure of the component subsets.

Details:

  1. Two different component subsets may share one or more components.
  2. In case multiple statements are true, select the strongest one.
w = ∏ k = 1 m ( 1 − ∏ i ∈ C k ( 1 − p i ) ) w = \prod_{k=1}^{m}\big(1-\prod_{i \in C_k}(1-p_i)\big) None of these w ≤ ∏ k = 1 m ( 1 − ∏ i ∈ C k ( 1 − p i ) ) w \leq \prod_{k=1}^{m}\big(1-\prod_{i \in C_k}(1-p_i)\big) w ≥ ∏ k = 1 m ( 1 − ∏ i ∈ C k ( 1 − p i ) ) w \geq \prod_{k=1}^{m}\big(1-\prod_{i \in C_k}(1-p_i)\big)

Abhishek Sinha by Abhishek Sinha , 34, India

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