In outdoor climbing, it is often necessary to build an anchor out of a set of cl

Question

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, 13 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. Find the probability that the anchor fails. Find the probability that it rained given that the anchor failed. Find the probability that piece B failed given that the anchor holds.

Explanation / Answer

Let

A, B, C = A, B, C fail, respectively
D = dry
R = rain

a)

Note that all 3 gears should fail.

Hence,

P(all fail) = P(D) P(A|D) P(B|D) P(C|D) + P(R) P(A|R) P(B|R) P(C|R)

= (1-0.20)*0.1*0.5*0.2 + 0.20*0.2*1*0.4

= 0.024 [ANSWER]

************************

b)

P(R|all fail) = P(R) P(A|R) P(B|R) P(C|R)/P(All failed)

= 0.20*0.2*1*0.4/0.024

= 0.666666667 [ANSWER]

*************************

c)

P(holds) = 1 - P(All fail) = 1 - 0.024 = 0.976

Hence,

P(B'|holds) = [P(D) P(B'|D) + P(R)P(B'|R)]/P(holds)

= ((1-0.20)*0.5 + 0.20*1)/0.976

= 0.614754098 [ANSWER]

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