Q5 BT: Card Game
Question 5 from the book “A Practical Guide to Quantitative Finance Interviews” by Zhou Xinfeg.
Card Game Problem Statement
A casino offers a card game using a normal deck of 52 cards. The rule is that you turn over two cards each time. For each pair, if both are black, they go to the dealer’s pile; if both are red, they go to your pile; if one black and one red, they are discarded. The process is repeated until you two go through all 52 cards. If you have more cards in your pile, you win $100; otherwise (including ties) you get nothing. The casino allows you to negotiate the price you want to pay for the game. How much would you be willing to pay to play this game?
Info processing
Card count is 52.
2 cards picked each time.
Both are black, dealer’s pile.
Both are red, your pile.
One red, one black, discard hand.
Solution and Explanation
Bring the number down to 10. Is there any one way to deal the cards so you win? There is none! The game always ends in a draw and your opponent wins.
10 cards (5r, 5b)
- 2b : (5r, 3b)
- 2r : (3r, 3b)
- 1b 1r: : (2r, 2b)
- 2b : (3r, 0b)
- 2r
Black: 4 Red: 4 Draw: .. You lose
How about if for all pickings, the pair picked is a single color till all of that color is exhausted from the deck?
Total Card count: 10 cards (5r, 5b)
| Picking Round | Black picked | Red picked | Red left | Black left |
|---|---|---|---|---|
| 1 | 2 | 0 | 5 | 3 |
| 2 | 2 | 0 | 5 | 1 |
| 3 | 1 | 1 | 4 | 0 |
| 4 | 0 | 2 | 2 | 0 |
| 5 | 0 | 2 | 0 | 0 |
| Total | 4 | 4 | nil | nil |
Total for black is a 4, total for red is a 4. It’ll always be that way regardless of the deck size, if there’s an equal number of each of black and red.
This is a rigged game. There’s always an equal number of red and black pairs. The game will always draw. The house always wins. I would not play it.
That’s it for this one.