2014 AIME I Problems/Problem 2

Revision as of 15:56, 14 March 2014 by Tanuagg13 (talk | contribs) (Problem 2)

Problem 2

An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is 0.58. Find $N$.

Solution

First, we find the probability both are blue, then the probability both are green, and add the two probabilities which equaling $0.58$. The probability both are blue is $\frac{4}{10}\cdot\frac{16}{16+N}$, and the probability both are green is $\frac{6}{10}\cdot\frac{N}{16+N}$, so \[\frac{4}{10}\cdot\frac{16}{16+N}+\frac{6}{10}\cdot\frac{N}{16+N}=\frac{29}{50}.\] Solving this equation, we get $N=144$.