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 Equal Chance (Posted on 2009-08-03)
Lana and Lois each had 15 red and 6 blue marbles. Each placed at least 1 red marble and at least 4 blue marbles into a bag (each had her own bag for this purpose). Each calculated the probability that if she drew out four marbles from her own bag that all four drawn would be blue. It turns out that both the probabilities are the same even though Lana had put more marbles in her bag than Lois had put in hers.

How many of each color had been placed in each bag?

 Submitted by Charlie No Rating Solution: (Hide) Lana had placed 4 red and 6 blue marbles in her bag. Lois had placed 3 red and 5 blue marbles in her bag. ```DEFDBL A-Z FOR red = 1 TO 15 FOR col = 4 TO 6 p = col / (col + red) * (col - 1) / (col + red - 1) * (col - 2) / (col + red - 2) * (col - 3) / (col + red - 3) PRINT red; col, PRINT USING "#.#######"; p NEXT NEXT ``` produces the following table (which, though, has since been sorted on the probability column): ``` 15 4 0.0002580 14 4 0.0003268 13 4 0.0004202 12 4 0.0005495 11 4 0.0007326 10 4 0.0009990 15 5 0.0010320 14 5 0.0012900 9 4 0.0013986 13 5 0.0016340 8 4 0.0020202 12 5 0.0021008 15 6 0.0025063 11 5 0.0027473 7 4 0.0030303 14 6 0.0030960 10 5 0.0036630 13 6 0.0038700 6 4 0.0047619 12 6 0.0049020 9 5 0.0049950 11 6 0.0063025 8 5 0.0069930 5 4 0.0079365 10 6 0.0082418 7 5 0.0101010 9 6 0.0109890 4 4 0.0142857 8 6 0.0149850 6 5 0.0151515 7 6 0.0209790 5 5 0.0238095 3 4 0.0285714 6 6 0.0303030 4 5 0.0396825 5 6 0.0454545 2 4 0.0666667 3 5 0.0714286 4 6 0.0714286 3 6 0.1190476 2 5 0.1428571 1 4 0.2000000 2 6 0.2142857 1 5 0.3333333 1 6 0.4285714 ``` Only the rows for 3 red, 5 blue and 4 red 6 blue contain the same probability. Based upon Enigma No. 1551, "The same chance", by Richard England, New Scientist, 27 June 2009.

 Subject Author Date Solution Brian Smith 2017-07-26 21:11:01 Hints ed bottemiller 2009-08-03 13:43:04

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