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