What is the probability of one more broken part in a system with 25 moving parts of which 7 have already broken and been replaced?

Question

This is a probability math problem. Past a certain number of broken moving parts occurring in a system with 25 moving parts, the probability of one more breaking actually starts to increase, not decrease.

Answer 1

Not sure if I am understanding the setting correctly. If all of the 7 broken parts have been replaced, the system is just like a brand new system to me. From the frequentist standing point, the probability of one more broken part is just the next broken part, which should be an independent event.

Answer 2

Each part in the system has a probability of failing that is associated with that part (a basic property of the part just like color). However, unlike color, typically part failure is a function of part usage and so it changes as the part is used. A typical part might have a higher failure rate initially (accounting for manufacturing defects) then settle into a low failure rate for most of it's life, then go up at the end of it's life until it fails.

One question to ask "Does a part failure mean that the system fails?" In other words, are all parts required to make the system function so that if any one fails, the system fails. I assume that what you'd really like to know is "What is the probability of the system of 25 moving parts failing given that 7 have already broken and been replaced and that all parts are required to make the system function?". To calculate this you would typically just multiply each part's reliability all together to determine the system reliability.

A website that has a good basic description of this is:

http://www.cqeweb.com/Chapters-HTML/Chap8_html/reliability.htm