Question

At the MAT 133 Casino, there is a curious game. Two cards are drawn, at the same time, from a
shuffled deck. The kings, queens, and jacks have value zero. Every other card’s value is its rank.
For example, the value of 3♣ is three and the value of J♥ is zero. To play this game, you must
pay 1$. The values of the cards are added together. If the total value is bigger than eighteen, then
you will receive M dollars. For what values of M is this game profitable for casino? You may use
that 52C2 = 1326

Answer 1

The maximum value can be 20 (Here we have considered the rank of Ace as 1)

To recieve M dollars, the values should be greater than 18 i.e 19 or 20

Number of ways in which cards can be drawn such that the sum is 20 = 4C2 = 6 (Both 10)

and the sum is 19 in 4C1*4C1 = 16 (One 10 and one 9)

Thus, total number of ways of selecting the cards in which you get sum greater than 18 = 22

Probability that you receive M dollars = 22/1326

The expected value of the game should be greater than 0 for the casino for the game to be be profitable

Thus, 1 - 22/1326*M should be greater than 0

-> M < 60.27

Thus, for a value of M less than or equal to $60, the game is profitable for casino