Consider a system that consists of independent components

MATHEMATICS · Multi-part · CBSE · Mathematics · Probability · Probability

Problem

Consider a system that consists of

n

independent components

C_1, C_2, \dots , C_n

. Let

E_i

be the event that component

C_i

functions satisfactorily during a mission of specified length. (i) State clearly what it means for the events

E_1,E_2,\dots,E_n

to be independent. (ii) If the system works only when all the

n

components work, derive an expression for the probability that the system works in terms of

P(E_i),\;i=1,2,\dots ,n

. (iii) If the system works provided at least one of the

n

components works, derive an expression for the probability that the system works in terms of

P(E_i),\;i=1,2,\dots ,n

. Show each logical step clearly.