There's this riddle: In a pub the owner is throwing a number of dice simultaneously. "I am trying to get one of each of the six faces", he says, "But it hasn't happened yet". "No", I said "You need at least four more dice to make the odds in favour of such a thing." How many dice does the owner have?
I'm confused because why can we not simply calculate the expected value of the number of times we need to roll a single dice in order to get all 6 values. If this was the case, then we can reduce this problem down to the coupon collectors problem, and see that the expected number of dice is 14.7.
Apparently, for the correct answer, we need to use inclusion/exclusion, and you ultimately get 13 dice.
Therefore, this is my question:
Why can we not use expected value, and why does using expected value get us a different answer?
Best Answer
If you don’t understand why something doesn’t work, think of something simpler where you’d also expect it to work and where it would be easier to understand why it doesn’t work.
Consider a five-sided die (made with finest Swiss craftsmanship to have equiprobable faces despite the lack of symmetry). You want to roll a number greater than $2$ (i.e. $3$, $4$ or $5$). How many dice do you need for the odds to be in favour of that? One. How many rolls of one die do you expect to need to get such a number? $\frac1{\frac35}=\frac53$, so the nearest integer is $2$.
It’s hard to explain why something doesn’t work when you haven’t said much about why you think it should work, but this example might help you clarify your thoughts on this. If you still don’t see why it doesn’t work, perhaps it will at least make it easier for you to explain why you think it should work, and that in turn would make it easier for us to explain why it doesn’t.