What is the expected payoff for this game?

Question

A carnival game involves drawing a card from a deck of 40 cards, replacing it, shuffling the deck, and drawing another card. Thirty of the cards are blank and ten are labeled 8 tokens. Your prize is the sum of the cards you draw. What is the expected payoff for this game?

Answer 1

 

  Probability of selecting the blank card is 30/40 = 3/4

 Probability of selecting the card with 8 stars is 10/40 = 1/4

 

  The sample space is    {( O,O), (O,X), (X,O), (X,X) } where O represents the blank card and X represent the card with 8 stars

 

    Their probabilities are 3/4*3/4 = 9/16

                                       3/4*1/4 = 3/16

                                       1/4*3/4 = 3/16

                                       1/4*1/4 = 1/16

   Note the total of these probabilities is 1, as it should

 

The expected value is then 9/16*0 + 3/16*8 + 3/16*8 + 1/16*16

                                       = 0 + 24/16 + 24/16 + 16/16

                                       = 64/16

                                          = 4