I'm encountering the following interesting math problem:
A device consists of 5 independently working blocks.
Each of them has a damage probability of 1/4. Upon damage of 1, 2 or 3 blocks, the probabilities for shutting the device down are respectively 1/5 , 2/5, 4/5.
If more blocks are damaged the device's probability of shutting down is 1 (100%). What is the expected number of damaged blocks? If the device has shut down, what is the probability exactly 2 blocks to be damaged?
What I tried: I tried combining the probabilities for the damaged probability blocks, and then dividing them, but I don't seem to do it properly.
I also thought of: P(A or B) = P(A) + P(B) - P(A and B), but still... to no avail.
Any idea how to proceed will be of use!