I came across this question while reading random questions on probability.
"One hundred doors, one dollar after each door. Roll a one-hundred dice for one hundred times. You can take the dollar after the door whose number is rolled out. What's the expectation? Why?"
My first instinct was to use Markov chains of some sorts, but that require a really large number of equations. I was wondering if anyone could tell me what I'm not looking at carefully or which direction I should follow.
P.S.: I have taken an introductory 400 level course in probability and ended up understanding a decent amount. I was wondering what more I could do to get deeper into probability. Thanks!
On a lot of these types of problems, you can take advantage of the fact that expectation is linear, i.e., $\mathbb{E}[X+Y] = \mathbb{E}[X]+\mathbb{E}[Y]$, whether or not $X$ and $Y$ are independent.
Then there's some cleverness involved in expressing a random variable as a linear combination of new random variables. Here, let $D$ be the total number of dollars you collect. Let $D_i$ be the total number of dollars you collect from door $i$ (so $D_i$ takes on values $0$ or $1$). Then you have $D = \sum_{i=1}^{100} D_i$, so $\mathbb{E}[D] = \sum_{i=1}^{100}\mathbb{E}[D_i]$.
Now calculating $\mathbb{E}[D_i]$ should be doable.