SOLUTION: Card Game. A game is set up as follows: All the diamonds are removed from a deck of cards, and these 13 cards are placed in a bag. The cards are mixed up, and then one card is chos

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Question 1142815: Card Game. A game is set up as follows: All the diamonds are removed from a deck of cards, and these 13 cards are placed in a bag. The cards are mixed up, and then one card is chosen at random (and then replaced). The player wins according to the following rules.
If the ace is drawn, the player loses $20. If a face card is drawn, the player wins $10 If any other card (2-10) is drawn, the player wins $2 How much should be charged to play this game in order for it to be fair?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The probabilities of drawing any one of the 13 cards are all the same. So the expected value for a play is the average of the values of each card.

The only losing card is the ace; the player loses $20 if that card is drawn.

Each of the 3 face cards wins $10; each of the remaining 9 cards wins $2.

So the average value of each draw is

%28%28-20%29%2B3%2810%29%2B9%282%29%29%2F13+=+%28-20%2B48%29%2F13+=+28%2F13

On average, the player will WIN (28/13) dollars; to be a fair game, the player must PAY (28/13) dollars for a game.

That of course is not possible, unless the game must be played in groups of 13 draws, with a cost of $28 for the 13 draws.