Mike and Bill play a card game with a standard deck of 52 cards. Mike selects a card from a well-shuffled deck andreceives A dollars from Bill if the card selected is a diamond and receives $1 from Bill if the card selected is an ace that is not adiamond. Otherwise, Mike pays Bill two dollars. Determine the value of A if the game is to be fair.

Question
Asked Oct 16, 2019

Mike and Bill play a card game with a standard deck of 52 cards. Mike selects a card from a well-shuffled deck and
receives A dollars from Bill if the card selected is a diamond and receives $1 from Bill if the card selected is an ace that is not a
diamond. Otherwise, Mike pays Bill two dollars. Determine the value of A if the game is to be fair.

check_circleExpert Solution
Step 1

The total number of cards are 52. Mike will gain ‘A’ dollars if the card selected is diamond.

Probability of gaining ‘A’ dollars can be calculated as:

Total diamond cards
P(A)=
T otal cards
13
52
1
Again he gain 1 dollar if the card selected is an ace and it is not a diamond. Then probability of
gaining 1 dollar can be calculated as:
Total ace cards exc luding diamond
P(1)
Total cards
52
He will pay 2 dollars otherwise. Then probability of paying 2 dollar can be calculated as:
Total cards excluding diamonds and ace
P(2)
Total cards
36
52
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Total diamond cards P(A)= T otal cards 13 52 1 Again he gain 1 dollar if the card selected is an ace and it is not a diamond. Then probability of gaining 1 dollar can be calculated as: Total ace cards exc luding diamond P(1) Total cards 52 He will pay 2 dollars otherwise. Then probability of paying 2 dollar can be calculated as: Total cards excluding diamonds and ace P(2) Total cards 36 52

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Step 2

The probability distributio...

Gain
1
-2
A
36
1
Probability
4 5252
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Gain 1 -2 A 36 1 Probability 4 5252

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