Question

In this game, you will play in pairs with a single deck of cards (face cards removed). Aces count as ones and all numbered cards count at face value. Players take turns flipping over two cards and finding the product. Regardless of who flipped the cards, Player 1 always gets one point if the product is even, and Player 2 always gets one point if the product is odd. Continue playing until one player reaches 20 points. Would you rather be Player 1 or Player 2? Do you think this is a fair game? If you think it is not, how could you alter the rules to make it fair?

Answer 1

The total number of ways of getting a product of 2 cards = 9 * 8 = 72

The product of 2 cards will be even if

(a) Both cards are even or (b) one card is even and the other card is odd. Therefore in 9 cards of which 5 are odd and 4 are even

(a) No of ways of getting both even cards = 4 * 3 = 12 (1card we choose 1 out of 4 even cards, and second card choose 1 out of the remaining 3 even cards)

(b) number of ways of getting 1 card even and the other odd = 2 * 5 * 4 = 40 (either choose 1 even card and then an odd card or choose an odd card then an even card)

Total number of ways = 40 + 12 = 52

The product of 2 cards are odd only if both the cards are odd. Total number of ways = 5 * 4 = 20

I would rather be Player 1 as there is a higher chance of getting a pair of cards which will give me a product which is even.

This is definitely not be a fair game as the number of options of getting an even number is more than that of getting an odd number.

To make it a fair game make 2 changes in the rules:

1) Remove Ace as 1. You will now have 4 even numbers (2,4,6,8) and 4 odd numbers(3,5,7,9).

2) Player 1 gets a point if both cards are even, and player 2 gets a point if both cards are odd.