Question

A system consisting of two components is subject to a series of shocks that arrive according to a Poisson process with a rate of one per day. Half the shocks cause no damage, one in ten damages component 1 alone, one in five damages component 2 alone, and the rest damage both components. Note that if only one component is functioning, and a shock occurs that would damage both components, the functioning component is damaged. Compute the probability that the at least one component is still functioning after 5 days.

Answer 1

P(at least one component working after 5 days) = 1 - P(no component working after 5 days)

= 1 - [P(shock damages both Comp1 & Comp2) x P(at least one shock occurred in the five day period)]

-------------------------------------------------------(1)

Given,

P(shock damages no component)=0.5

P(shock damages Comp1)=0.1

P(shock damages Comp2)=0.2

Therefore, P(shock damages both Comp1 & Comp2)=1-(0.5+0.1+0.2)=0.2 ----------------------------------(2)

Now using Poisson Distribution, we calculate P(at least one shock occurred in the five day period)

= 5 shocks per 5 day interval

then P(no shock occured in the 5 day interval) = (50 x e-5)/0! =0.006738

Therefore, P(at least one shock occurred in the five day period) = 1- 0.006738 =0.993262 ---------------------(3)

Substituting values from (2) and (3) into (1), we get

P(at least one component working after 5 days) = 1 - 0.2 x 0.993262

= 0.8013 Ans