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Wins 2 - wins all (Posted on 2010-05-06) Difficulty: 3 of 5
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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Solution solution | Comment 2 of 5 |

By the symmetry of the situation, A and B have equal likelihood of winning, with C having the remaining probability. So if we find the probability that C will win, we can split the remaining probability in two, for A and B.

In a given set of three games, there is a 1/2 probability that the winner of the first game (A or B) will also win the second game and therefore the prize. In the remaining 1/2 probability (that C wins the second game) there is the further, conditional, probability of 1/2 that C would also win the next one, for a probability of 1/4 of C winning on this round. In the 1/4 remaining probability of no winner thus far, there is a 1/2 conditional probability that the winner of the third game (who must be A or B, as C would have had to have won the second game and lost the third to proceed this far) will be the winner of the fourth game, which is the first game of the next round. So that's an additional 1/8 probability that it's not C, leaving 1/8 undecided.

The between-round difference (if the match continues) between the winner of the last game in the preceding round and the winner of the first game of the next round does not affect the calculations in the subsequent round, so the cycle continues from here.

So in any given round, C has a 1/4 probability of winning, and 1/8 that the match will continue to another round (counting the round to begin actually on the second game, as that's what determines the first of our win types).

In situations like this, the ultimate probabilities are proportional to the probabilities within one round. In this case, the within-round probabilities are 5/16, 5/16 and 4/16, or a ratio of 5:5:4. Therefore overall, A and B each have a probability of 5/14 of getting the prize and C has a probability of 4/14. These are approximately .3571428571428572 and .2857142857142857, respectively.

A simulation verifies these probabilities:

DECLARE SUB play3 (round!)
DIM SHARED s$, ct(3), tct

DO
play3 1
LOOP

SUB play3 (round)
 STATIC pl$
 SELECT CASE round MOD 3
  CASE 1
   pl$ = "ab"
  CASE 2
   pl$ = RIGHT$(s$, 1) + "c"
  CASE 0
   IF LEFT$(pl$, 1) = "a" THEN
    pl$ = "b" + "c"
   ELSE
    pl$ = "a" + "c"
   END IF
 END SELECT
 r = INT(RND(1) * 2 + 1)
 w$ = MID$(pl$, r, 1)
 IF round = 1 THEN prev$ = "":  ELSE prev$ = RIGHT$(s$, 1)
 s$ = s$ + w$
 IF prev$ = w$ THEN
   ct(INSTR("abc", w$)) = ct(INSTR("abc", w$)) + 1
   tct = tct + 1
   PRINT s$; TAB(15); ct(1); ct(2); ct(3), ct(1) / tct; ct(2) / tct; ct(3) / tct
   s$ = ""
 ELSE
   play3 round + 1
 END IF
 
END SUB

The last few result lines, when the program was interrupted, were:

this round's         cumulative
winners        a wins  b wins  c wins       p(a)      p(b)     p(c)
------------   ------  ------  ------      ------------------------------
aa             969973  970264  776622      .35702  .3571271  .2858529
bb             969973  970265  776622      .3570199  .3571273  .2858528
bcc            969973  970265  776623      .3570197  .3571272  .2858531
acc            969973  970265  776624      .3570196  .3571271  .2858533
bcaa           969974  970265  776624      .3570198  .357127  .2858532
aa             969975  970265  776624      .3570201  .3571268  .2858531
aa             969976  970265  776624      .3570203  .3571267  .285853
aa             969977  970265  776624      .3570206  .3571266  .2858529
aa             969978  970265  776624      .3570208  .3571264  .2858528
acc            969978  970265  776625      .3570206  .3571263  .2858531
bb             969978  970266  776625      .3570205  .3571265  .2858529
bb             969978  970267  776625      .3570204  .3571268  .2858528
acbb           969978  970268  776625      .3570203  .357127  .2858527
bb             969978  970269  776625      .3570201  .3571272  .2858526
bcabcabcaa     969979  970269  776625      .3570204  .3571271  .2858525
aa             969980  970269  776625      .3570206  .357127  .2858524
bcc            969980  970269  776626      .3570205  .3571268  .2858527
acc            969980  970269  776627      .3570203  .3571267  .2858529
aa             969981  970269  776627      .3570206  .3571266  .2858528
bcc            969981  970269  776628      .3570204  .3571264  .2858531
aa             969982  970269  776628      .3570207  .3571263  .285853
bcabb          969982  970270  776628      .3570206  .3571266  .2858529
bcc            969982  970270  776629      .3570204  .3571264  .2858531
bb             969982  970271  776629      .3570203  .3571267  .2858531
bb             969982  970272  776629      .3570202  .3571269  .2858529
bb             969982  970273  776629      .35702  .3571271  .2858528
acbaa          969983  970273  776629      .3570203  .357127  .2858527
bb             969983  970274  776629      .3570201  .3571272  .2858526
acc            969983  970274  776630      .35702  .3571271  .2858529
acbb           969983  970275  776630      .3570199  .3571273  .2858528
bb             969983  970276  776630      .3570198  .3571276  .2858527
aa             969984  970276  776630      .35702  .3571275  .2858526
bcc            969984  970276  776631      .3570198  .3571273  .2858528
bb             969984  970277  776631      .3570197  .3571275  .2858527
acbb           969984  970278  776631      .3570196  .3571278  .2858526
bb             969984  970279  776631      .3570195  .357128  .2858525
bb             969984  970280  776631      .3570193  .3571283  .2858524
bcc            969984  970280  776632      .3570192  .3571281  .2858527
aa             969985  970280  776632      .3570194  .357128  .2858526
aa             969986  970280  776632      .3570197  .3571279  .2858525
aa             969987  970280  776632      .3570199  .3571277  .2858524
bcaa           969988  970280  776632      .3570201  .3571276  .2858523
bcabcaa        969989  970280  776632      .3570204  .3571275  .2858522
acbacc         969989  970280  776633      .3570202  .3571273  .2858524

  Posted by Charlie on 2010-05-06 13:46:30
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