Probability and words: An urn contains W white marbles and B black marbles....

PaulCarlock

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[FONT=&quot][FONT=&quot]An urn contains W white marbles and B black marbles. Two marbles are randomly and simultaneously drawn from the urn. (The two marbles might both be black, or might both be white, or might be of different colors.) If the probability that the marbles have the same color is 0.5 (i.e. both are white or both are black), then W and B must be ??????????? ?????????? numbers. (The first word has 11 letters, the second word has 10.)[/FONT][/FONT]
 
An urn contains W white marbles and B black marbles. Two marbles are randomly and simultaneously drawn from the urn. (The two marbles might both be black, or might both be white, or might be of different colors.) If the probability that the marbles have the same color is 0.5 (i.e. both are white or both are black), then W and B must be ??????????? ?????????? numbers. (The first word has 11 letters, the second word has 10.)

Interesting problem! But did you read the Read before posting message? We want you to show your work, so we can help you figure it out, not just give you an answer.

I've got the 11-letter word, but am not sure of the other. To find them, you'll have to take it step by step.

A good way to start, if you have no idea what to do, is to pick some numbers for W and B and see how you'd calculate the probability. Suppose there are, say, 3 white and 4 black marbles? What do you get?

Then you can do the same with variables and see what happens in general.
 
An interesting question with a perhaps unexpected answer that deserves a reply. (I can answer only because I have seen this on another forum!)

W white marbles
B black marbles
N (= W + B) total marbles

(I do not have the skills to write equations neatly, hope this is clear)

(a) Probability of drawing 2 white marbles = (W/N) x ((W - 1)/(N - 1))
(b) Probability of drawing 2 black marbles = ((N - W)/N) x ((N - W - 1)/(N - 1))
(a) + (b) = 0.5
(W/N) x ((W - 1)/(N - 1)) + ((N - W)/N) x ((N - W - 1)/(N - 1)) = 0.5
(W x (W - 1)) + ((N - W) x (N - W - 1)) = N(N - 1) / 2
2W^2 - 2W + 2N^2 - 2NW - 2N - 2NW + 2W^2 + 2W = N^2 - N
4W^2 - 4NW + (N^2 - N) = 0

Solving as a quadratic for W
W = (4N +/- sqrt(16N^2 - 4x4(N^2 - N))) / 2x4
W = (4N +/- sqrt(16N^2 - 16N^2 + 16N)) / 8
W = (4N +/- sqrt(16N)) / 8
W = (N +/- sqrt(N)) / 2

We are dealing with positive integers so N (the total number of marbles) must be a perfect square.
The number of one colour will be (N - sqrt(N))/2, the other (N + sqrt(N))/2
An infinite series of numbers give answers
Total Colour1 Colour2
4 1 3
9 3 6
16 6 10
25 10 15
36 15 21

The series 1, 3, 6, 10, 15, 21 ... ... is that of the triangular numbers
So your answer is : W and B must be consecutive triangular numbers.
 
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