2002 AMC 12P Problems/Problem 15

Problem

There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find $|P_s-P_d|.$

$\text{(A) }0 \qquad \text{(B) }\frac{1}{2002} \qquad \text{(C) }\frac{1}{2001} \qquad \text{(D) }\frac {2}{2001} \qquad \text{(E) }\frac{1}{1000}$

Solution

First we find the value of $P_s$ . Note that whatever color we choose on our first marble, there are exactly $1000$ of $2001$ marbles remaining that match that color. Therefore, $P_s = \frac {1000}{2001}$ .

Now we find the value of $P_d$ . Again, the actual color of the first marble does not matter, since there are always exactly $1001$ of $2001$ marbles remaining that match that color. Therefore, $P_d = \frac{1001}{2001}$ .

The value of $|P_s - P_d|$ is therefore $\frac {|1000-1001|}{2001} = \boxed {\text{(C) }\frac{1}{2001}}$ .

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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2002 AMC 12P Problems/Problem 15

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