Difference between revisions of "2019 AMC 10A Problems/Problem 22"

Revision as of 17:25, 9 February 2019

The following problem is from both the 2019 AMC 10A #22 and 2019 AMC 12A #20, so both problems redirect to this page.

Problem

Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$ . Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \tfrac{1}{2}$ ?

$\textbf{(A)} \frac{1}{3} \qquad \textbf{(B)} \frac{7}{16} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{9}{16} \qquad \textbf{(E)} \frac{2}{3}$

Solution

There are several cases depending on what the first coin flip is when determining $x$ and what the first coin flip is when determining $y$ .

The four cases are:

$x$ is either $0$ or $1$ and $y$ is either $0$ or $1$
$x$ is either $0$ or $1$ and $y$ is chosen from the interval $[0,1]$
$x$ is is chosen from the interval $[0,1]$ and $y$ is either $0$ or $1$
$x$ is is chosen from the interval $[0,1]$ and $y$ is chosen from the interval $[0,1]$

Each case has a $\frac{1}{4}$ chance of occurring.

For case 1, we need $x$ and $y$ to be different. Therefore, the probability for success in case 1 is $\frac{1}{2}$ .

For case 2, if $x$ is 0, we need $y$ to be in the interval $(\frac{1}{2}, 1]$ . If $x$ is 1, we need $y$ to be in the interval $[0, \frac{1}{2})$ . Regardless of what $x$ is, the probability for success for case 2 is $\frac{1}{2}$ .

By symmetry, case 3 has the same success rate as case 2.

For case 4, we must use geometric probability because there are an infinite number of pairs $(x, y)$ that can be selected, whether they satisfy the inequality or not. Graphing $|x-y| > \tfrac{1}{2}$ , gives us the following picture where the shaded area is the set of all the points that fulfill the inequality:

$[asy] filldraw((0,0)--(0,1)--(1/2,1)--(0,1/2)--cycle,black+linewidth(1)); filldraw((0,0)--(1,0)--(1,1/2)--(1/2,0)--cycle,black+linewidth(1)); draw((0,0)--(0,1)--(1,1)--(1,0)--cycle); [/asy]$

The shaded area is $\frac{1}{4}$ , which means the probability for success for case 4 is $\frac{1}{4}$ .

Adding up the success rates from each case, we get:

$(\frac{1}{4}) \cdot (\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{4}) = \boxed{\textbf{(B) }\frac{7}{16}}$ .

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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 10 Problems and Solutions

2019 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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.

@@ Line 21: / Line 21: @@
 For case 1, we need <math>x</math> and <math>y</math> to be different. Therefore, the probability for success in case 1 is <math>\frac{1}{2}</math>.
-For case 2, if x is 0, we need y to be in the interval <math>(\frac{1}{2}, 1]</math>. If x is 1, we need y to be in the interval <math>[0, \frac{1}{2})</math>. Regardless of what x is, the probability for success for case 2 is <math>\frac{1}{2}</math>.
+For case 2, if <math>x</math> is 0, we need <math>y</math> to be in the interval <math>(\frac{1}{2}, 1]</math>. If <math>x</math> is 1, we need <math>y</math> to be in the interval <math>[0, \frac{1}{2})</math>. Regardless of what <math>x</math> is, the probability for success for case 2 is <math>\frac{1}{2}</math>.
 By symmetry, case 3 has the same success rate as case 2.

Page

Toolbox

Search

Difference between revisions of "2019 AMC 10A Problems/Problem 22"

Revision as of 17:25, 9 February 2019

Problem

Solution

See Also