Difference between revisions of "2007 AMC 12B Problems/Problem 13"
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| − | ==Problem | + | == Problem == |
A traffic light runs repeatedly through the following cycle: green for <math>30</math> seconds, then yellow for <math>3</math> seconds, and then red for <math>30</math> seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? | A traffic light runs repeatedly through the following cycle: green for <math>30</math> seconds, then yellow for <math>3</math> seconds, and then red for <math>30</math> seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? | ||
| − | <math>\mathrm {(A)} \frac{1}{63} | + | <math>\mathrm{(A)}\ \frac{1}{63} \qquad \mathrm{(B)}\ \frac{1}{21} \qquad \mathrm{(C)}\ \frac{1}{10} \qquad \mathrm{(D)}\ \frac{1}{7} \qquad \mathrm{(E)}\ \frac{1}{3}</math> |
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| + | == Solution == | ||
The traffic light runs through a <math>63</math> second cycle. | The traffic light runs through a <math>63</math> second cycle. | ||
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<math>\frac{9}{63} = \frac{1}{7} \Rightarrow \mathrm{(D)}</math> | <math>\frac{9}{63} = \frac{1}{7} \Rightarrow \mathrm{(D)}</math> | ||
| − | ==See Also== | + | == See Also == |
{{AMC12 box|year=2007|ab=B|num-b=12|num-a=14}} | {{AMC12 box|year=2007|ab=B|num-b=12|num-a=14}} | ||
| + | |||
| + | [[Category:Introductory Combinatorics Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 00:03, 19 October 2020
Problem
A traffic light runs repeatedly through the following cycle: green for 
seconds, then yellow for 
seconds, and then red for seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?
Solution
The traffic light runs through a 
second cycle.
Letting 
reference the moment it turns green, the light changes at three different times: 
, 
, and 
This means that the light will change if the beginning of Leah's interval lies in
![$[27,30]$](http://latex.artofproblemsolving.com/f/e/d/fed247ea7f8c97f0f1ded70f012592db772169a4.png)
, ![$[30,33]$](http://latex.artofproblemsolving.com/e/c/3/ec3d9f67ec329c02168e5579090fe14df62de947.png)
or ![$[60,63]$](http://latex.artofproblemsolving.com/b/8/3/b83cfa74157bcba9f5ccdeaba2515a13d5bb7acd.png)
This gives a total of 
seconds out of
See Also
| 2007 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 12 |
Followed by Problem 14 |
| 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. 
