In an interview I received the follow question: We have 3 cards face down, and we give each card in a deck of 52 a numeric score ( A = 1, 2=2, .... , J=11, Q=12, K = 13). The interviewer asked me to find the expected sum of these 3 cards, I approximated that each card has an expected value of (1+2+...+13)/13 = 7, and so the sum is approximately going to be 21 (or adjusted down slightly to 20.5). Then the game begins as follows.
You begin with $1000. I will quote a price to play this game, and you are able to either buy x units, or sell me x units, then we turn over the cards and you realize your profit or loss. For example:
Lets say you are given a price of 25, clearly this is greater than 21, so I would choose to sell (Short) 10 units at that price. The cards turn out to have a sum of 23, so I make a profit of (25-23)*10 = 20. So my total money now is 1020, and so on.
We played several rounds of this, and I was just wondering what is an appropriate strategy for this game? I tended to increase the units I would buy when quoted a very low price, and increase the number I would sell when quoted a very high price (relative to 21), but now that I think about it, is there a more sophisticated way of approaching this problem?