Difference between revisions of "2015 AMC 12B Problems/Problem 17"
(→Solution) |
(→Problem) |
||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
| − | An unfair coin lands on heads with a probability of <math>\tfrac{1}{4}</math>. When tossed <math>n>1</math> times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the | + | An unfair coin lands on heads with a probability of <math>\tfrac{1}{4}</math>. When tossed <math>n>1</math> times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the valu of <math>n</math>? |
<math>\textbf{(A)}\; 5 \qquad\textbf{(B)}\; 8 \qquad\textbf{(C)}\; 10 \qquad\textbf{(D)}\; 11 \qquad\textbf{(E)}\; 13</math> | <math>\textbf{(A)}\; 5 \qquad\textbf{(B)}\; 8 \qquad\textbf{(C)}\; 10 \qquad\textbf{(D)}\; 11 \qquad\textbf{(E)}\; 13</math> | ||
Revision as of 21:23, 25 December 2018
Contents
Problem
An unfair coin lands on heads with a probability of 
. When tossed 
times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the valu of 
?
Solution
When tossed times, the probability of getting exactly 2 heads and the rest tails is
![\[\dbinom{n}{2} {\left( \frac{1}{4} \right)}^2 {\left( \frac{3}{4} \right) }^{n-2}.\]](http://latex.artofproblemsolving.com/f/c/6/fc6405c602dee5d12cb017541fb674a4f3bbb8e8.png)
Similarly, the probability of getting exactly 3 heads is
![\[\dbinom{n}{3}{\left( \frac{1}{4} \right)}^3 {\left( \frac{3}{4} \right) }^{n-3}.\]](http://latex.artofproblemsolving.com/9/2/c/92c95422814935e2a1744e40bb8ff525c4db41a3.png)
Now set the two probabilities equal to each other and solve for :
![\[\dbinom{n}{2}{\left( \frac{1}{4} \right)}^2 {\left( \frac{3}{4} \right) }^{n-2}=\dbinom{n}{3}{\left( \frac{1}{4} \right)}^3 {\left( \frac{3}{4} \right) }^{n-3}\]](http://latex.artofproblemsolving.com/d/e/e/deed9cb1861f787abde4c577294d240a24de5890.png)
![\[\frac{n(n-1)}{2!} \cdot \frac{3}{4} = \frac{n(n-1)(n-2)}{3!} \cdot \frac{1}{4}\]](http://latex.artofproblemsolving.com/0/7/f/07f1b2a26ad277631bd46c1314db7092bcd944c6.png)
![\[3 = \frac{n-2}{3}\]](http://latex.artofproblemsolving.com/5/d/4/5d414de3ecbe06f23e64517629c32e1a0023e6d3.png)
![\[n-2 = 9\]](http://latex.artofproblemsolving.com/1/c/d/1cd6ebfe15d5b338b44028394dc9b9543fe488f7.png)
![\[n = \fbox{\textbf{(D)}\; 11}\]](http://latex.artofproblemsolving.com/c/2/7/c27f5da250dd157f54a2a7e3c2e4eb2eb7e4ed48.png)
Note: the original problem did not specify , so 
was a solution, but this was fixed in the Wiki problem text so that the answer would make sense. — @adihaya (talk) 15:23, 19 February 2016 (EST)
Solution 2
Bash it out with the answer choices! (not really a rigorous solution)
See Also
| 2015 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 16 |
Followed by Problem 18 |
| 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. 
