The water for a small town is provided from two sources A and B. In any given dr

Question

The water for a small town is provided from two sources A and B. In any given dry season, there is a 0.01 probability that Source A will not be able to provide the expected amount of water needed. The corresponding probability for Source B is 0.02. If Source A is not able to meet the demand, there is a 0.3 probability that B will not be able to either. If only one source cannot meet the demand, the probability of water shortage in town will be 0.40; whereas, if both sources cannot meet the demand, the probability of water shortage in the town will be 0.90.

a. Compute the probability of water shortage in the town in the next season.

b. If there is a shortage, what is the probability that only Source A is causing it?

Explanation / Answer

Here we are given that:

P( A not able to meet ) = 0.01 and P( B not able to meet ) = 0.02

Also, we are given tht given A is not able to meet the demand, there is a 0.3 probability that B will not be able to either.

Therefore, P( B not able to meet | A not able to meet ) = 0.3

Using Bayes theorem, we get:

P( both not able to meet the demand ) = P( B not able to meet | A not able to meet ) P(A not able to meet )

P( both not able to meet the demand ) = 0.3*0.01 = 0.003

P( only one of them not able to meet ) = P(A not able to meet ) + P(B not able to meet ) - 2P( both not able to meet the demand ) = 0.01 + 0.02 - 2*0.003 = 0.024

Also we are given that P( water shortage | one of them not able to meet ) = 0.4 and

P( water shortage | Both not able to meet ) = 0.9

a) Now using law of total probability, we get:

P( water shortage ) = P( water shortage | Both not able to meet ) P( both not able to meet the demand ) + P( only one of them not able to meet )P( water shortage | one of them not able to meet )

P( water shortage ) = 0.9*0.003 + 0.4*0.024 = 0.0123

Therefore the probability of having a water shortage is equal to 0.0123

b) P( only A is not able to meet ) = P( A is not able to meet ) - P( both are not able to meet ) = 0.01 - 0.003 = 0.007

Now using bayes theorem, we get:

P( only A is not able to meet | water shortage ) = P( water shortage | Only A is not able to meet ) P( only A is not able to meet ) / P( water shortage )

P( only A is not able to meet | water shortage ) = 0.4*0.007 / 0.0123 = 0.2276

Therefore 0.2276 is the required probability here.

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