Number of Marbles in a Bag Containing Black and White Marbles

Let
b = number of black marbles
w = number of white marbles
 

Note:
The probability of removing n marbles (1 marble at a time) in succession without replacement is equal to the probability of removing n marbles at once.
 

2 Marbles are Removed
P_{First 2 are white} = 1/3
 

3 Marbles are Removed
P_{First 2 are white}   P_{Third is white} = 1/6
(1/3) P_{Third is white} = 1/6
P_{Third is white} = 1/2
 

Therefore, the number of black and white marbles are equal after removing 2 white marbles.

b = w - 2
total number of marbles = b + w
total number of marbles = 2w - 2

 

First Draw White, Second Draw White
$\dfrac{w}{2w - 2} \cdot \dfrac{w - 1}{2w - 3} = \dfrac{1}{3}$

$3w(w - 1) = (2w - 2)(2w - 3)$

$3w^2 - 3w = 4w^2 - 10w + 6$

$w^2 - 7w + 6 = 0$

$(w - 1)(w - 6) = 0$

$w = 1 ~ \text{ and } ~ 6$
 

Use w = 6 marbles
$b = 6 - 2$

$b = 4 ~ \text{ black marbles}$   ←   answer