Imagine a LONG bike with a total of 4 wheels - 2 in the front and 2 in the back.
For the bike to work at all, at least 1 front wheel and 1 back wheel must be operational.
All 4 tires are the same and each, individually, has probability 'p' of failing.
What is the probability that I will have a working bike? (meaning, that at least 1 tire is working up front, and at least 1 is working in the back).
When you want to know the probability that more than one probability will occur, you multiply the probabilities. To find out if either one or another of events will occur, you add the probabilities.
Thus you would use the following formula:
1 - (pF1 + pF2)(pR1 + pR2), where pF1 is the probability for one front tire failing.
However, since the probability is the same for each tire, the formula would be:
1 - (p + p)(p + p) = 1 - (2p)(2p) = 1 - 4p
I hope this helps. Thanks for asking.
Pardon my typo.
1 - (2p)(2p) = 1 - 4p^2
That is p squared.
Sorry for the omission.
No problem! Thank you for catching that. The correct formula should be:
1 - (2p)(2p) = 1 - 4p^2
So the probability of having a working bike, where at least 1 tire is working up front and at least 1 tire is working in the back, is 1 - 4p^2.