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 Marble Game (Posted on 2004-11-01)
You're playing a game. You start with a box with one black marble and one white marble, and you sample twice with replacement. If you select the white marble both times, you win. If you select the black marble either time, you add another black marble and try again. On each round, you sample twice with replacement, winning if you select the white marble twice, otherwise adding another black marble and moving on to the next round.

What is the probability that you eventually win? Equivalently, if P(n) is the probability that you win on or before round n, what is the limit of P(n) as n -> infinity?

 See The Solution Submitted by Brian Smith Rating: 4.2000 (5 votes)

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 some values | Comment 4 of 12 |

For the first few n:

`   n prob(exactly n) P(n) -- on or before n   1 0.250000000000 0.250000000000   2 0.083333333333 0.333333333333   3 0.041666666667 0.375000000000   4 0.025000000000 0.400000000000   5 0.016666666667 0.416666666667   6 0.011904761905 0.428571428571   7 0.008928571429 0.437500000000   8 0.006944444444 0.444444444444   9 0.005555555556 0.450000000000  10 0.004545454545 0.454545454545  11 0.003787878788 0.458333333333  12 0.003205128205 0.461538461538  13 0.002747252747 0.464285714286  14 0.002380952381 0.466666666667  15 0.002083333333 0.468750000000  16 0.001838235294 0.470588235294  17 0.001633986928 0.472222222222  18 0.001461988304 0.473684210526  19 0.001315789474 0.475000000000  20 0.001190476190 0.476190476190  21 0.001082251082 0.477272727273  22 0.000988142292 0.478260869565  23 0.000905797101 0.479166666667  24 0.000833333333 0.480000000000  25 0.000769230769 0.480769230769  26 0.000712250712 0.481481481481  27 0.000661375661 0.482142857143  28 0.000615763547 0.482758620690  29 0.000574712644 0.483333333333  30 0.000537634409 0.483870967742  31 0.000504032258 0.484375000000  32 0.000473484848 0.484848484848  33 0.000445632799 0.485294117647  34 0.000420168067 0.485714285714  35 0.000396825397 0.486111111111  36 0.000375375375 0.486486486486  37 0.000355618777 0.486842105263  38 0.000337381916 0.487179487179  39 0.000320512821 0.487500000000  40 0.000304878049 0.487804878049`

continued on, it goes through...

`152618 0.000000000021 0.499996723868152619 0.000000000021 0.499996723889152620 0.000000000021 0.499996723911152621 0.000000000021 0.499996723932152622 0.000000000021 0.499996723954152623 0.000000000021 0.499996723975152624 0.000000000021 0.499996723997`

so it does look like .5 is the answer.  But most people will weary of the game before the probability gets this close to .5.

DEFDBL A-Z
CLS
black = 1
cLoss = 1
DO
w = cLoss * 1 / ((black + 1) * (black + 1))
cWin = cWin + w
cLoss = cLoss - w
PRINT USING "######"; ct + 1;
PRINT USING " #.############"; w; cWin
black = black + 1
ct = ct + 1
' IF ct MOD 40 = 0 THEN DO: LOOP UNTIL INKEY\$ > ""
LOOP

 Posted by Charlie on 2004-11-01 14:25:51

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