Equation (1):

Pr(X = x) = A(6.75x - x ),

with x {1,2,3,4,5}

Equation (2):

Pr(X = x | Y = y) = ((0.988 - 0.008x)^(y-1)) (0.012 + 0.008x)

with y {1,2,3,....}

1. Using the law of total probability and equations (1) and (2), find the chance that a given circuit (selected at random) fails on precisely its fourth day of use, Pr(Y = 4).

2. Using Bayes theorem and your result for Question 1, find the probability that a device which failed on its fourth day of use had precisely four of its internal circuit components activated, Pr(X = 4|Y = 4).

Background Info:

Mass Production

To save on costs, a certain circuit is mass-produced, and then adjusted before shipping to different manufactures to tailor it to their needs. Specifically, five different sub-components of the circuit can be switched on or off, according to what the different manufacturers buying the circuit require. Historical production data for these circuits suggests that the best model for the prob- ability that a manufacturer requests a circuit with X of these sub-components switched to on is given by the probability mass function

Pr(X = x) = A(6.75x x2), (equation 1)

with x {1,2,...,5}.

A range of similar devices each use one of these circuits, but the expected lifetime of these devices varies with the number of active sub-components used in the circuit. Specifically, the baseline failure rate of the circuit during any given day is 1.2%, and then this is increased by 0.8% for each sub-component of the circuit that is set to on. This results in the following conditional probability for the device failing on day Y, given the number of components that are on, X:

Pr(Y = y|X = x) = ((0.988 0.008x)^y1) (0.012 + 0.008x). (equation 2)

where y {1,2,3,...}.