Here is an interesting problem...

[Steve]

A system consists of two main parts. Part “A” warranty ends at 10 years. Part “B” warranty ends at 20 years.

It is assumed that if a part “A” fails after 10 years, the entire unit will be replaced since the customer’s cost of replacing part “A” is very nearly the cost of a complete, new unit.

How would one calculate the probability that part “B” will fail (and therefore be replaced under warranty) between years 10 and 20 given that part “A” has NOT failed in the same time period?

I understand that this is a conditional probability problem, but the particulars for solving the problem are sticky.

[Arai.M]

Are the failures of A and B independent events? if that is the case (and getting rid of the conditional probabilities for now), you have
P(X|Y) = P(X)*P(Y) for independent events

now replace X with probability of B failing between 10 and 20 and Y with probability of A not failing between 10 and 20.

You have now

P(X)*P(Y) = P(B fails t<20 | B hasn't failed t<10)
* P(A doesn't fail t<20 | A hasn't failed t<10)
= 1 - P(B doesn't fail t<20 | B hasn't failed t<10)
* P(A doesn't fail t<20 | A hasn't failed t<10)

These are now conditional reliabilities, where R(T+t|T) = R(T+t)/R(T) which will depend on the distributions that describe these 2 parts.
If these are dependent events (e.g. if when A fails, B is more likely to fail) then it becomes sticky because you have to know how to describe that dependency.

[Steve]

Did you mean P(XandY) = P(X)*P(Y) not P(X|Y) since P(X|Y) = P(X) for independent events?

[Arai.M]

My bad, you are right. The rest should apply though...

[Rui Assis]

Allow me to contribute to this thread with a different approach. I prefer simulation in this case, in order to skip over formulae and use only logic. I add an EXCEL application with an example where part B may fail during the period 10 to 20 years due to both an inherent failure or due to a failure originated in part A. The distributions of failure of parts A and B as well as the probability that part A may cause a failure in part B are assumed to be known. Hope this will be of help.

Regards,

[etcottle]

Your spreadsheet is essentially a Mone Carlo simulation, which I also use when the math gives me a headache. There is a less time-consuming approach, though, that doesn't require hitting "F9" 1000 times. Weibull++ has a Monte Carlo data generator (the "dice" icon on the toolbar). If I were doing this problem, and I knew the failure distributions of A and B, including the conditional probability of failure of B, P(B|A), I would generate 1000 or so random failures of each case. I would then save them to an Excel spreadsheet, and do all 1000 iterations at once.

[Rui Assis]

I also have Weibull++ but I enjoy to develop my own applications whenever I can, as it is a good chance to understand in full depth what is actually going on. To perform the repetitions and get frequency distributions and confidence intervals of the output variables, I use to run a macro in Excel with my own code which is very fast.

Rui

[Arai.M]

Since I happen to be very keen on simulation simulation as well, I though I would give yet another alternative. I am attaching a Reno file. You can download a demo version at http://www.reliasoft.com/reno/evaluation.htm. In that file, you will see a flowchart where failures for both A and B are generated and a sequence of conditional blocks (or if statements) arrive at the solution of the probability of B failing under warranty given A is not under warranty anymore and hasn't failed. Enjoy!