One final round quant interview problem. Three attempts. No hints, and no higher-or-lower — at this level that would just be a search.
A token starts at 0 and moves +1 or -1 with equal probability each step. It stops on reaching -3 or +5. On average, how many steps does that take?
Yesterday #2
A well shuffled 52 card deck is dealt one card at a time. On average, at which position does the first ace appear?
Answer: 10.6 position in the deck
Ignore the aces for a moment and look at the other 48 cards. The four aces cut them into five blocks: the cards before the first ace, the cards between the first and the second, and so on down to the cards after the fourth.
No block is special, so by symmetry all five have the same expected size. They share 48 cards, so each averages 48/5 = 9.6.
The first ace sits immediately after the first block, so its expected position is 9.6 + 1 = 10.6.
Both common wrong answers skip the symmetry. 9.6 is the block itself, which counts the cards before the ace rather than the position of the ace. 13 is 52/4, which would hold if the aces split the deck into four equal parts, but a shuffle does not place them that regularly.