You are to play the following game of cards. Cards are worth their face value, J

Question

You are to play the following game of cards. Cards are worth their face value,
Jacks, Queens and Kings are also worth 10 and Aces are worth 11. You pull precisely two cards
from a standard deck and if their sum is at least 20, then you win the game and are given a
prize of cash. It costs you $10 to play the game. At least how much should the prize be in
order to make the odds of winning or breaking even in your favor? Note that if the sum of your
cards is less than 20 then you lose your $10.

Explanation / Answer

Let X be the value on the two cards. two aces: P(X=22) = (4/52)(3/51) = 1/221 one ace, one face: P(X=21) = (4/52)(12/51) + (12/52)(4/51) = 8/221 one ace, one nine or two face: P(X=20) = (4/52)(4/51)+(4/52)(4/51)+(12/52)(11/51) = 53/663 so P(X at least 20) = 1/221 + 8/221 + 53/663 = 80/663 let A be amount of prize to break even: (A-10)(80/663) + (-10)(583/663) = 0 so (A-10)(80) = (10)(583) ==> A-10 = 583/8 ==> A = 10+583/8 = $85.875 The prize needs to be at least $85.88

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