I am trying to solve a question in probability which goes as follows:
There is a series of light bulbs on the street. Each light bulb may burn out with probability p every day and be out of use, independently of the other light bulbs.
a) What is the expected value of the number of days for which the first N bulbs will last?
b) What is the expected value for the number of light bulbs that will last for K days?
My answer for (a) is as follows:
For each $i\in N$ can denote random variable $Y_i$ to signify the number of days since $i-1$ bulbs burnt out until $i$ bulbs burnt out. We then get for each $i$ that $Y_i$ is geometrically distributed (dependent on $i$) and sum all these random variables to get the total number of days the bulbs lasted, lets denote it $Y$. We can then calculate the sum of the expected values of these variables and get $E[Y]$.
But as for (b), I don't know how to approach it - if I know that the bulbs have to last K days, what kind of random variables can I create so that their sum, or maybe maximum, I don't know, gives us the amount of time they survived together? And how can we get the expected value for that?