perplexus.info :: Probability : Wins 2

Wins 2 - wins all (Posted on 2010-05-06)

Three backgammon players of equal game skills compete for a prize.
The prize will be awarded to the winner of two games in a row.
A and B, following a drawing, play the first game, then the winner will face C.
Next game, if needed, will be by C and the player who lost the 1st game and so on.
Determine a priori chances of winning for each of the 3 players, assuming Lady Luck treats them without discrimination.

Rem: There are no draws in backgammon.

No Solution Yet Submitted by Ady TZIDON

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analytical solution

| Comment 1 of 5

listing who can win in each round:

round 1: nobody

round 2: a,b

round 3: c (2 ways)

round 4: a,b

round 5: a,b

round 6: c (2 ways)

round 7: a,b

round 8: a,b

round 9: c (2 ways)

....

so it can be seen that A and B have equal chances of winning so we need only compute C's chances. The probability of C winning is equal to

2*(1/2)^3 + 2*(1/2)^6 + 2*(1/3)^9 + .....

which is equal to 2/7

thus the odds for A and B are (1- (2/7))/2 = 5/14

which can also be confirmed by computing the infinite sum

(1/4) + (1/2)^4 + (1/2)^5 + (1/2)^7 + (1/2)^8 + ....

thus confirming that somebody will eventually win

thus the odds are

A: 5/14 or approximately 35.7%

B: 5/14 or approximately 35.7%

C: 2/7 or approximately 28.6%

Posted by Daniel on 2010-05-06 12:46:18

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