After taking TINFO 240, Christopher of the Soprano crime family is excited to tell Tony about a new gambling game he has created that will generate funds for the family. The ules are a player draws one card from a standard 52-card deck of playing cards. If the player picks a heart, the player will win $10. If the player picks a face card that is not a heart (King Queen Jack), the player wins $8. Picking a face card heart counts the same as picking a heart. If a player picks any other card, they lose $6. Tony listens and then tell Christopher he is an idiot again. Who is right, is this game a money maker for the Sopranos? Explain.

To calculate whether or not a betting game is good in the long run, we use expected value (which is a measure of what the outcome of the game is on average). When playing the game there are 3 outcomes for the player: Either the player wins $10, wins $8, or loses $6.

Expected value of playing the game for the player is computed by adding each of those outcomes, but each multiplied by how likely it is to happen (a weighted sum).

The player wins $10 if he/she picks a heart from a standard deck of cards. There are 13 hearts in the deck, so the probability of doing so is 13/52.

The player wins $8 if he/she picks a face card which is not a heart from the deck. There are 12 total face cards, and excluding the heart ones, there are 9. So the probability of doing so is 9/52.

The player loses $6 if he/she picks any other card, which is picking any other card besides hearts, and face cards. There are 30 other cards besides hearts and face cards, so the probability of doing so is 30/52.

Therefore the expected value of playing the game for the player is

E[ playing the game ] = [tex] 10\cdot \frac{13}{52}+8\cdot \frac{9}{52}-6\cdot \frac{30}{52}=0.423[/tex]

Which means on average the player is expected to win $0.423 per game. Which means the game is not a money maker for the Sopranos.