Difference between revisions of "2014 AIME I Problems/Problem 2"
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+ | == Problem 2 == | ||
+ | An urn contains <math>4</math> green balls and <math>6</math> blue balls. A second urn contains <math>16</math> green balls and <math>N</math> blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is <math>0.58</math>. Find <math>N</math>. | ||
+ | |||
+ | == Solution == | ||
+ | First, we find the probability both are green, then the probability both are blue, and add the two probabilities. The sum should be equal to <math>0.58</math>. | ||
+ | |||
+ | The probability both are green is <math>\frac{4}{10}\cdot\frac{16}{16+N}</math>, and the probability both are blue is <math>\frac{6}{10}\cdot\frac{N}{16+N}</math>, so | ||
+ | <cmath> \frac{4}{10}\cdot\frac{16}{16+N}+\frac{6}{10}\cdot\frac{N}{16+N}=\frac{29}{50}</cmath> | ||
+ | Solving this equation, | ||
+ | <cmath>20\left(\frac{16}{16+N}\right)+30\left(\frac{N}{16+N}\right)=29</cmath> | ||
+ | Multiplying both sides by <math>16+N</math>, we get | ||
+ | |||
+ | <cmath>\begin{align*} | ||
+ | 20\cdot16+30\cdot N&=29(16+N)\\ | ||
+ | 320+30N&=464+29N\\ | ||
+ | N&=\boxed{144} | ||
+ | \end{align*}</cmath> | ||
+ | |||
+ | == See also == | ||
+ | {{AIME box|year=2014|n=I|num-b=1|num-a=3}} | ||
+ | {{MAA Notice}} | ||
+ | [[Category:Intermediate Probability Problems]] |
Latest revision as of 20:12, 15 January 2022
Problem 2
An urn contains green balls and
blue balls. A second urn contains
green balls and
blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is
. Find
.
Solution
First, we find the probability both are green, then the probability both are blue, and add the two probabilities. The sum should be equal to .
The probability both are green is , and the probability both are blue is
, so
Solving this equation,
Multiplying both sides by
, we get
See also
2014 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.