# Probability of an event that is conditonial on other events

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Hi. I got a problem which is related to Probabilities Theory.

Suppose that we have a system that consists of, for example, 5 parts a, b, c, d and e. And also we have these informations: If a fails, the system will fail with probability of 50%. If b fails, the system will fail with probability of 40%. And If c, d and e don't fail, the system won't fail with probability of 35%, 40% and 45% respectively.

Now this is the question:

What is the probability of the system not being failed, if a and b are failed and c, d, e, aren't. Please explain the method, too.

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It depends on whether or not the failures of the various parts are independent or not. If they are independent, the the probability of not failure is simply the product of not failure of the individual parts under the given condition. (.5x.6x.35x.4x.45)

If they are dependent you need to know the relationships.

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They are independent and you are right. Thank you so much. Somtimes simple and abvious things seem more complex.

Wait a moment. Now I think it is wrong. In this case the probability of not failure is 0.0189 and the probability of failure is (.5x.4x.65x.6x.55)=0.0429. It is wrong, isn't it?

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They are independent and you are right. Thank you so much. Somtimes simple and abvious things seem more complex.

Wait a moment. Now I think it is wrong. In this case the probability of not failure is 0.0189 and the probability of failure is (.5x.4x.65x.6x.55)=0.0429. It is wrong, isn't it?

Your failure prob. is wrong. Not failure means ALL not fail, so multiply. Failure means at least one fails, so you need to add all possible combinations - easier to calculate 1 - prob(not fail).

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