probability problem independent events

I have a problem about probability and independent events, could anybody help me to solve this problem?:
a. The Pogo Thorhead rocket will function properly only if all five of its major systems operate simultaneously. The systems are labeled A,B,C,D and E. System failures are independent, and each system has a probability of failure of 1/3. Given the Thorhead fails, determine the probability that system A is solely at fault.
b. The Pogo Thorhead II is an improved configuration using the same five systems, each still with a probability of failure of 1/3. The Thorhead II will fails if system A fails or if (at least) ant two systems fail. Alas, the Thorhead II also fails. Determine the probability that system A is solely at fault.

Regards

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Originally Posted by rbenito

a. The Pogo Thorhead rocket will function properly only if all five of its major systems operate simultaneously. The systems are labeled A,B,C,D and E. System failures are independent, and each system has a probability of failure of 1/3. Given the Thorhead fails, determine the probability that system A is solely at fault.

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Let a denote A works, and -a denote A fails and similary for the other systems. Also let t denote that Thorhead works and -t that it fails.

P(-a|-t) = P(-a)P(-t|-a)/P(-t)

this is Bayes theorem where P(u|v) is the conditional probability of u given that v has occured. By definition here P(-t|-a)=1, so we have:

P(-a|-t) = P(-a)/P(-t) = (1/3)/P(-t)

Note P(-t) = 1 - P(no systems fail) = 1 - (2/3)^5, so:

P(-a|-t) = (1/3)/(1-(2/3)^5)

RonL

Quote:

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Originally Posted by rbenito

b. The Pogo Thorhead II is an improved configuration using the same five systems, each still with a probability of failure of 1/3. The Thorhead II will fails if system A fails or if (at least) ant two systems fail. Alas, the Thorhead II also fails. Determine the probability that system A is solely at fault.

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Use the same notation as before but now let -a* denote the event that
only system A fails. Then as before we have:

P(-a*|-t) = P(-a*)P(-t|-a*)/P(-t),

and as before p(-t|-a*)=1, so:

P(-a*|-t) = P(-a*)/P(-t).

I will leave it to you from here.

RonL