A bag holds white and black marbles, with 10 more white than black.

probability/same.color.half New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

A bag holds white and black marbles, with 10 more white than black.
If two marbles are drawn (without replacement), the chance that they
are the same colour is 1/2.  How many marbles of each colour are there?

[ Editor: problem statement reconstructed from the solution posting. ]

Show solution

The algebraic computations for this problem follow:

Given:          White (W) = Black (B) + 10
                P(2B) + P(2W) = .5

Assumption:     Drawing without replacement

Probabilities:  P(2B)  = [B / (B + W)] * [(B - 1) / (B + W - 1)]
                P(2W)  = [W / (B + W)] * [(W - 1) / (B + W - 1)]

-----------------------------------------------------------------

     B * (B - 1) + W * (W - 1)   =  .5   =  1
(1)  -------------------------              -
       (B + W) * (B + W - 1)                2


(2)  2 * [B * (B - 1) + (B + 10) * (B + 9)] = (B + B + 10) * (B + B + 9)

(3)  4 * B^2 + 36 * B + 180  =  4 * B^2 + 38 * B + 90

(4)  90 = 2 * B

(5)  B = 45; ergo W= 55.

-- Dick Adams <rdadams@ubmail.ubalt.edu>