Difference between revisions of "2020 AMC 12B Problems/Problem 16"
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After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color? | After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color? | ||
<math>\textbf{(A)}\ \frac{1}{6} \qquad\textbf{(B)}\ \frac{1}{5} \qquad\textbf{(C)}\ \frac{1}{4} \qquad\textbf{(D)}\ \frac{1}{3} \qquad\textbf{(E)}\ \frac{1}{2}</math> | <math>\textbf{(A)}\ \frac{1}{6} \qquad\textbf{(B)}\ \frac{1}{5} \qquad\textbf{(C)}\ \frac{1}{4} \qquad\textbf{(D)}\ \frac{1}{3} \qquad\textbf{(E)}\ \frac{1}{2}</math> | ||
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| + | ==Solution== | ||
| + | Let the probability that the urn ends up with more red balls be denoted <math>P(R)</math>. Since this is equal to the probability there are more blue balls, the probability there are equal amounts is <math>1-2P(R)</math>. <math>P(R) =</math> the probability no more blues are chosen plus the probability only 1 more blue is chosen. The first case, <math>P(\text{no more blues}) = \frac{1}{2}*\frac{2}{3}*\frac{3}{4}*\frac{4}{5}=\frac{1}{5}</math>. | ||
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| + | The second case, <math>P(\text{1 more blue}) = 4*\frac{1*1*2*3}{2*3*4*5} = \frac{1}{5}</math>. Thus, the answer is <math>1-2(\frac{1}{5}+\frac{1}{5})=1-\frac{4}{5}=\boxed{\textbf{(B)}\ \frac{1}{5}}</math>. | ||
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| + | ~JHawk0224 | ||
| + | |||
| + | ==See Also== | ||
| + | |||
| + | {{AMC12 box|year=2020|ab=B|num-b=15|num-a=17}} | ||
| + | {{MAA Notice}} | ||
Revision as of 21:47, 7 February 2020
Problem
An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn.
After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
Solution
Let the probability that the urn ends up with more red balls be denoted 
. Since this is equal to the probability there are more blue balls, the probability there are equal amounts is 
. 
the probability no more blues are chosen plus the probability only 1 more blue is chosen. The first case, 
.
The second case, 
. Thus, the answer is 
.
~JHawk0224
See Also
| 2020 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 15 |
Followed by Problem 17 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
