Decision making

Question

Which of the following 3 games of chance should you play?

1. Draw a card from a standard deck. If it is an ace, you win $5; if it is red, you win $1 ( a red ace wins $6); anything else, you lose $2.

2. Draw a card from a standard deck. If it is a club or a face card, you win $2.50; anything else, you lose $1.

3. Draw 2 cards, without replacement, from a standard deck. If they are of the same suit, you win $5; if it is a pair, you win $10; anything else, you lose $2.

Answer 1

The way to figure this out is to calculate the expected value, and play the game with the highest expected value.  The expected value is the sum of all the probabilities of winning or losing times the respective amounts.  Losing bets are counted as negative amounts.

 

First game:

 

Probability of drawing a red ace from a standard deck is 2/52 = 1/26 = 0.0384165

Probability of drawing a red card that is not a red ace is 24/52 = 6/13 = 0.4615384

Probability of drawing a black ace from a standard deck is 2/52 = 1/26 = 0.0384165

Probability of drawing a black card that is not an ace is 24/52 = 6/13 = 0.4615384

 

Expected value = 0.0384165*$6.00 + 0.4615384*$1.00 + 0.0384165*$5.00 - 0.4615384*$2.00 =

 

0.230499 + 0.4615384 + .1920825 - 0.9230768 = -0.0389569

 

Second game:

 

Probability of drawing a club or a face card is 22/52 = 0.4230769

Probability of drawing anything else = 30/52 = 0.576923

 

Expected value = 0.4230769*$2.50 - 0.576923*$1.00 = 1.0576922 - 0.576923 = 0.4807692

 

Third game:

 

Probability of drawing two cards in a row from the same suit = 13/52 *  12/13 = 0.2307692

Probability of drawing a pair = 4/52 * 3/4 = 0.0576923

Probability of drawing anything else = 0.7115385

 

Expected value = 1.153846 + 0.576923 -1.423077 =  0.307692

 

SECOND GAME has the highest expected value and is positive, so that is the game to play.

Answer 2

Find the expected value of the three games and choose the game with the largest expected value.