Difference between revisions of "2016 AMC 8 Problems/Problem 21"
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+ | ==Problem== | ||
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A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn? | A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn? | ||
<math>\textbf{(A) }\dfrac{3}{10}\qquad\textbf{(B) }\dfrac{2}{5}\qquad\textbf{(C) }\dfrac{1}{2}\qquad\textbf{(D) }\dfrac{3}{5}\qquad \textbf{(E) }\dfrac{7}{10}</math> | <math>\textbf{(A) }\dfrac{3}{10}\qquad\textbf{(B) }\dfrac{2}{5}\qquad\textbf{(C) }\dfrac{1}{2}\qquad\textbf{(D) }\dfrac{3}{5}\qquad \textbf{(E) }\dfrac{7}{10}</math> | ||
− | == | + | ==Solutions== |
− | + | ===Solution 1=== | |
− | |||
− | ==Solution 1== | ||
We put five chips randomly in order, and then pick the chips from the left to the right. To find the number of ways to rearrange the three red chips and two green chips, we solve for <math>\binom{5}{2} = 10</math>. However, we notice that whenever the last chip we draw is red, we pick both green chips before we pick the last (red) chip. Similarly, when the last chip is green, we pick all three red chips before the last (green) chip. This means that the last chip must be green in all the situations that work. This means we are left with finding the number of ways to rearrange three red chips and one green chip, which is <math>\binom{4}{3} = 4</math>. Because a green chip will be last <math>4</math> out of the <math>10</math> situations, our answer is <math>\boxed{\textbf{(B) } \frac{2}{5}}</math>. | We put five chips randomly in order, and then pick the chips from the left to the right. To find the number of ways to rearrange the three red chips and two green chips, we solve for <math>\binom{5}{2} = 10</math>. However, we notice that whenever the last chip we draw is red, we pick both green chips before we pick the last (red) chip. Similarly, when the last chip is green, we pick all three red chips before the last (green) chip. This means that the last chip must be green in all the situations that work. This means we are left with finding the number of ways to rearrange three red chips and one green chip, which is <math>\binom{4}{3} = 4</math>. Because a green chip will be last <math>4</math> out of the <math>10</math> situations, our answer is <math>\boxed{\textbf{(B) } \frac{2}{5}}</math>. | ||
− | ==Solution 2== | + | ===Solution 2=== |
− | There are two ways of ending the game, either you picked out all the red chips or you picked out all the green chips. We can pick out <math>3</math> red chips, <math>3</math> red chips and <math>1</math> green chip, <math>2</math> green chips, <math>2</math> green chips and <math>1</math> red chip, and <math>2</math> green chips and <math>2</math> red chips. Because order is important in this problem, there are <math>1+4+1+3+6=15</math> ways to pick out the chip. But we noticed that if you pick out the three red chips before you pick out the green chip, the game ends. So we need to subtract cases like that to get the total number of ways a game could end, which <math>15-5=10</math>. Out of the 10 ways to end the game, 4 of them ends with a | + | There are two ways of ending the game, either you picked out all the red chips or you picked out all the green chips. We can pick out <math>3</math> red chips, <math>3</math> red chips and <math>1</math> green chip, <math>2</math> green chips, <math>2</math> green chips and <math>1</math> red chip, and <math>2</math> green chips and <math>2</math> red chips. Because order is important in this problem, there are <math>1+4+1+3+6=15</math> ways to pick out the chip. But we noticed that if you pick out the three red chips before you pick out the green chip, the game ends. So we need to subtract cases like that to get the total number of ways a game could end, which <math>15-5=10</math>. Out of the 10 ways to end the game, 4 of them ends with a green chip. The answer is <math>\frac{4}{10} = \frac{2}{5}</math>, or <math>\boxed{\textbf{(B) } \frac{2}{5}}</math>. |
− | ==Solution 3== | + | ===Solution 3=== |
− | Assume that after you draw the three red chips in a row without drawing both green chips, you continue drawing for the next turn. The last/fifth chip that is drawn must be a green chip because if both green chips were drawn before, we would've already completed the game. So technically, the problem is asking for the probability that the "fifth draw" is a green chip. This probability is symmetric to the probability that the first chip drawn is green, which is <math>\frac{2}{5}</math>. | + | Assume that after you draw the three red chips in a row without drawing both green chips, you continue drawing for the next turn. The last/fifth chip that is drawn must be a green chip because if both green chips were drawn before, we would've already completed the game. So technically, the problem is asking for the probability that the "fifth draw" is a green chip. This probability is symmetric to the probability that the first chip drawn is green, which is <math>\frac{2}{5}</math>. So the probability is <math>\boxed{\textbf{(B) } \frac{2}{5}}</math>. |
Note: This problem is almost identical to 2001 AMC 10 #23. | Note: This problem is almost identical to 2001 AMC 10 #23. | ||
==Video Solution== | ==Video Solution== | ||
− | + | ||
+ | https://youtu.be/fTtUAtfWKyQ - Happytwin | ||
+ | |||
https://www.youtube.com/watch?v=w0y-JuRQvDc&feature=youtu.be | https://www.youtube.com/watch?v=w0y-JuRQvDc&feature=youtu.be | ||
+ | |||
+ | https://youtu.be/OOdK-nOzaII?t=1452 | ||
https://youtu.be/m834gDVyPJM | https://youtu.be/m834gDVyPJM | ||
− | + | ==See Also== | |
{{AMC8 box|year=2016|num-b=20|num-a=22}} | {{AMC8 box|year=2016|num-b=20|num-a=22}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 15:09, 2 January 2022
Contents
Problem
A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?
Solutions
Solution 1
We put five chips randomly in order, and then pick the chips from the left to the right. To find the number of ways to rearrange the three red chips and two green chips, we solve for . However, we notice that whenever the last chip we draw is red, we pick both green chips before we pick the last (red) chip. Similarly, when the last chip is green, we pick all three red chips before the last (green) chip. This means that the last chip must be green in all the situations that work. This means we are left with finding the number of ways to rearrange three red chips and one green chip, which is . Because a green chip will be last out of the situations, our answer is .
Solution 2
There are two ways of ending the game, either you picked out all the red chips or you picked out all the green chips. We can pick out red chips, red chips and green chip, green chips, green chips and red chip, and green chips and red chips. Because order is important in this problem, there are ways to pick out the chip. But we noticed that if you pick out the three red chips before you pick out the green chip, the game ends. So we need to subtract cases like that to get the total number of ways a game could end, which . Out of the 10 ways to end the game, 4 of them ends with a green chip. The answer is , or .
Solution 3
Assume that after you draw the three red chips in a row without drawing both green chips, you continue drawing for the next turn. The last/fifth chip that is drawn must be a green chip because if both green chips were drawn before, we would've already completed the game. So technically, the problem is asking for the probability that the "fifth draw" is a green chip. This probability is symmetric to the probability that the first chip drawn is green, which is . So the probability is .
Note: This problem is almost identical to 2001 AMC 10 #23.
Video Solution
https://youtu.be/fTtUAtfWKyQ - Happytwin
https://www.youtube.com/watch?v=w0y-JuRQvDc&feature=youtu.be
https://youtu.be/OOdK-nOzaII?t=1452
See Also
2016 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.