Black and White Marbles

Each of two boxes contains 20 marbles, and each marble is either black or white. The total number of black marbles is different from the total number of white marbles. One marble is drawn at random from each box. The probability that both marbles are white is .21. What is the probability that both are black?
Source: NCTM Mathematics Teacher, November 2006

SOLUTION
Let P(x,y) represent the probability of drawing marbles x from the first box and y from the second box. Drawing a marble from one box is an independent event from drawing a marble from another box. Let a represent the number of white marbles in the first box and b the number of white marbles in the second box.
P(white,white)=P(white)\times P(white)
.21=(a/20)(b/20)
21/100=(ab)/400
84/400=(ab)/400
ab=84
The possible factors that make 84 are
1\times 84
2\times 42
4\times 21
6\times 14
7\times 12
The first three are not possible because 84,42,21 are all greater than 20 marbles.
6\times 14 is not possible because that would make the number of white marbles equal to the number of black marbles
6 white +14 white=14 black +6 black.
The only solution is a=7 and b=12 or vice-versa a=12 or b=6.
P(black,black)=P(black)\times P(black)
=(20-7)/20\times (20-12)/20
=(13/20)\times (8/20)
=104/400
=.26

Answer: 0.26

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About mvtrinh

Retired high school math teacher.
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