In outdoor climbing it is often necessary to build an anchor

In outdoor climbing, it is often necessary to build an anchor out of a set of climbing gear. For safety, one should place at least three pieces of gear to create the anchor. Suppose we have built an anchor out of three pieces of gear, pieces A, B and C. The anchor fails only when A, B and C fail simultaneously. Given that it is a dry day, the probabilities that A, B or C fail are 0.1,0.5 and 0.2, respectively. On the other hand, if it is raining these probabilities of failure all double. The forecast for the day suggests the probability of rain is 20%. Given the weather, you may assume that the pieces of gear fail (or hold) independently.

Solution

Prob of anchor fail on dry day = 0.1 * 0.5 *0.2 = 0.01

so, Prob of anchor hold on dry day = 0.99

prob of B failed given that anchor hold = 0.8 * prob of B failure and anchor hold / prob of anchor hold

                                                             = 0.8 * (0.9 * 0.5 * 0.8 + 0.9 * 0.5 * 0.2 + 0.1 * 0.5 * 0.8) / 0.99

                                                             = 0.8 * 0.49 / 0.99 = 0.396   (ANS)