Difference between revisions of "2004 AIME II Problems/Problem 2"
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== Solution == | == Solution == | ||
| − | {{ | + | The [[probability]] that Terry picks two red candies is <math>\frac{10 * 9}{20 * 19} = \frac{9}{38}</math><!--<math>\frac{{10}\choose {2}}{{20}\choose {2}} = \frac{9}{38}</math>-->, and the probability that Mary picks two red candies is <math>\frac{7*8}{18*17} = \frac{28}{153}</math> (the same goes for the blue candies). The probability that Terry picks two different candies is <math>\frac{2*10*10}{20*19} = \frac{10}{19}</math><!--<math>\frac{{10}\choose {1} {10}\choose{1}}{{20}\choose {2}} = \frac{10}{19}</math>-->, and the probability that Mary picks two red candies is <math>\frac{2*9*9}{18*17} = \frac{18}{34}</math><!--<math>\frac{{9}\choose {1}{9}\choose {1}}{{18}\choose {2}} = \frac{18}{34}</math>-->. Thus, the probability becomes: |
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| + | :<math>2 * \frac{9}{38} * \frac{28}{153} + \frac{10}{19} * \frac{9}{17}</math> | ||
| + | :<math>= \frac{118}{323}</math> | ||
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| + | Therefore, the answer is <math>441</math>. | ||
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== See also == | == See also == | ||
| − | + | {{AIME box|year=2004|num-b=1|num-a=3|n=II}} | |
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Revision as of 19:14, 12 February 2007
Problem
A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is 
where 
and 
are relatively prime positive integers, find 
Solution
The probability that Terry picks two red candies is 
, and the probability that Mary picks two red candies is 
(the same goes for the blue candies). The probability that Terry picks two different candies is 
, and the probability that Mary picks two red candies is 
. Thus, the probability becomes:
Therefore, the answer is 
.
See also
| 2004 AIME II (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 | ||
