Life of a Crash Absorber
The Crooked Mountain Expressway has many bridge abutments with crash-absorbing barriers. Each year, one-third of these barriers are damaged in collisions and must be replaced or repaired. There is no apparent pattern to which bridge abutments are hit; rather, it appears to be random. If a crash absorber survives 5 years, it is replaced under regular maintenance. What is the probability distribution for the life of a crash absorber?

Question

Accepted Answer

This mathematical model assumes randomness of collisions over the years. Thus, if a crash absorber survives to the beginning of a year, there is a one-third chance of being hit during the year. For example, a crash absorber can be hit in year 2 only if it was not hit in year 1, and the one-third chance of a hit comes true in year 2. Thus, two equations form the probability model:

P(hit in year t) = P(not hit by year t − 1) · (1/3)

P(not hit by year t) = P(not hit by year t − 1) − P(hit in year t)

So, for the first year, the P(hit) = 1/3 and P(not hit yet) = 1 − 1/3 = 2/3. For the second year, the P(hit) = (2/3) · (1/3) = 2/9, and the P(not hit yet) = 2/3 − 2/9 = 4/9. This continues, as tabulated below, for 5 years. Finally, the probability that the crash absorber was not hit at all is found by subtracting the probabilities that it is hit from 1 (axiom 3). The distribution is shown graphically in Exhibit 18.1. The two 5-year outcomes could be combined (P = .1975).

P(not hit yet)	P(hit)	Year
.667	.333	1
.444	.222	2
.296	.148	3
.1975	.0988	4
.1317	.0658	5
Replaced anyway	.1317	5

Question 18.2: Life of a Crash Absorber The Crooked Mountain Expressway has...

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