Expectation of a game?

Question

A gambler draws a card from an ordinary deck of cards. If it is a black card, she wins $2. If it is a red card between or including 3 and 7, she wins $10. If it is a red face card, she wins $25, and if it is a black jack, she wins an additional $100. If it is any other card, she wins nothing. Find the expectation of the game.

Answer 1

To work this problem, we need to figure the probability of each payoff.

 

1. Probability of getting any black card except for the jacks is 24/52, this has a payoff of $2.

2. Probability of getting  any red card between 3 and 7 is 10/52, there are 5 cards between 3 and 7 and two suits of red,this has a payoff of $10.

3. Probability of getting a red face card is 6/52, as there are 3 face cards and two suits of red, this has a payoff of $25.

4. Probability of getting black jack is 2/52,this has a payoff of $102.

 

The expected value is the probability times the payoff

 

Expectation =  $2 * 24/52 + $10 * 10/52 + $25 * 6/52 + $102 * 2/52 = $9.6538