explain this please During its use a component may encounter one of two environm

Question

explain this please

During its use a component may encounter one of two environments: a normal environment, which is met with probability 0.98, and a severe one, with probability 0.02. In a normal environment the probability of a component failing is 0.03 whereas in a severe one this probability is 0.52.

Find: (a) The probability of component failure.

(b) The probability that a severe environment was encountered, given that the component has failed.

Write correct answers to 3 significant figures.

Explanation / Answer

Here we are given that:

P( normal ) = 0.98 and P( severe ) = 0.02

Also, we are given that:

P( failure | normal ) = 0.03 and P( failure | severe ) = 0.52

a) Using law of total probability, we get:

P( failure ) = P( failure | normal )P( normal ) + P( failure | severe )P( severe )

P( failure ) = 0.03*0.98 + 0.52*0.02 = 0.0398

Therefore 0.0398 is the required probability here.

b) Using bayes theorem, the probability that the environment is severe given that component failed is computed as:

P( severe | failed ) = P( failed | severe )P(severe ) / P( failed )

P( severe | failed ) = 0.52*0.02 / 0.0398 = 0.2613

Therefore 0.2613 is the required probability here.

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