John and Paul, who loved to gamble aganst each other, played a game that was fai

Question

John and Paul, who loved to gamble aganst each other, played a game that was fair to both. John was given a box with five marbles, marked 1, 1, 1, 3, 3. Paul was given a box with five marbles, marked 1, 2, 2, 2, 3. Each drew a marble from his own box. Had the two drawn the same number, no money would have changed hands, and had John drawn the higher number, whatever marbles they drew, Paul would have paid him $300. As it turned out, Paul drew the higher number.
How much did John pay Paul, assuming John's payment to Paul would have been the same whatever marbles they drew? Show your work.
(Hint: what are the possible scenarios favorable to each?)

Explanation / Answer

John was given a box with five marbles, marked 1, 1, 1, 3, 3. Paul was given a box with five marbles, marked 1, 2, 2, 2, 3

Let us list all the possiblities

Let the first number be Johns and the second Pauls

11 - tie

12 - paul wins

12 - paul wins

12- paul wins

13- paul wins

x3

____________

31 -john wins

32 -john wins

32 -john wins

32 -john wins

33 -john wins

x2

We need to find the possiblities : 15+ 10 = 25 total outcomes

Possiblity John wins 10/25

Possiblity Paul wins 12/25

To make the game fair if Paul wins he should get paid less because his odds are better

10/25*300 = 12/25*x

x = 250 dollars (John pays Paul)

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