Probability Distribution

Question

A card game is set up as follows:  The 13 hearts are placed in a bag and the player draws 1 card at random.  If they choose the ace, they lose $20.  If they choose the jack, queen or king, they win $10 and if they choose a numbered card, they win $2.  How much should be charged in order for this game to be fair?

Answer 1

draw probability outcome

ace 1/13 -20$ -x

j,q,k 3/13 +10$ - x

2-10 9/13 +2$ - x

x = amount of charge per bet

Fair when expected value is zero.

Expected value = sum of probability * outcome, then solve for x.

1/13(-20-x) + 3/13(10-x) + 9/13(2-x) = 0

-20/13 - 1/13x + 30/13 - 3/13x + 18/13 - 9/13x = 0

28/13 - 13/13x = 0

28/13 = x

x = 2.15