What is the expected time to roll all $1$ through $6$ on a six-sided die?
Recently I came across the above question. When I tried to solve the above question, I thought of $E(X + Y) = E(X) + E(Y)$. Implementing the above method to solve this question, I got the answer as $36$, which is definitely wrong. Can someone tell me what I am doing wrong? My solution is:
$$E(\text{all six}) = E(1) + E(2) + \ldots + E(6) = 6 + 6 + \ldots + 6 = 36$$
You are missing that those six events are not independent. If you needed to a roll a 1 through 6 in that order, your answer would be appropriate. But whatever you roll on the first throw is a number you don't need to throw again, so the analysis is going to be different.