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Equal Chance (Posted on 2009-08-03) Difficulty: 3 of 5
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    
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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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionBrian Smith2017-07-26 21:11:01
Hintsed bottemiller2009-08-03 13:43:04
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