Difference between revisions of "2016 AMC 12A Problems/Problem 23"

Revision as of 17:26, 16 November 2022

Problem

Three numbers in the interval $\left[0,1\right]$ are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?

$\textbf{(A) }\frac16\qquad\textbf{(B) }\frac13\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac23\qquad\textbf{(E) }\frac56$

Solution

Solution 1: Super WLOG

WLOG assume $a$ is the largest. Then, $b+c>a$ , meaning the solution is $\boxed{\textbf{(C)}\;1/2}$ , as shown in the graph below. $[asy] pair A = (0,0); pair B = (1,0); pair C = (1,1); pair D = (0,1); pair E = (0,0); draw(A--B--C--D--cycle); draw(B--D,dashed); fill(B--D--C--cycle,gray); label("$0$",A,SW); label("$a$",B,S); label("$a$",D,W); label("$y$",(0,.5),W); label("$x$",(.5,0),S); label("$x+y>a$",(5/7,5/7)); [/asy]$

Solution 2: Conditional Probability

WLOG, let the largest of the three numbers drawn be $a>0$ . Then the other two numbers are drawn uniformly and independently from the interval $[0,a]$ . The probability that their sum is greater than $a$ is $\boxed{\textbf{(C)}\;1/2.}$

Solution 3: Calculus

When $a>b$ , consider two cases:

1) $0<a<\frac{1}{2}$ , then $\int_{0}^{\frac{1}{2}} \int_{0}^{a}2b \,\text{d}b\,\text{d}a=\frac{1}{24}$

2) $\frac{1}{2}<a<1$ , then $\int_{\frac{1}{2}}^{1} \left(\int_{0}^{1-a}2b \,\text{d}b + \int_{1-a}^{a}1+b-a \,db\right)\text{d}a=\frac{5}{24}$

$a<b$ is the same. Thus the answer is $\frac{1}{2}$ .

Solution 4: Geometry

The probability of this occurring is the volume of the corresponding region within a $1 \times 1 \times 1$ cube, where each point $(x,y,z)$ corresponds to a choice of values for each of $x, y,$ and $z$ . The region where, WLOG, side $z$ is too long, $z\geq x+y$ , is a pyramid with a base of area $\frac{1}{2}$ and height $1$ , so its volume is $\frac{\frac{1}{2}\cdot 1}{3}=\frac{1}{6}$ . Accounting for the corresponding cases in $x$ and $y$ multiplies our answer by $3$ , so we have excluded a total volume of $\frac{1}{2}$ from the space of possible probabilities. Subtracting this from $1$ leaves us with a final answer of $\frac{1}{2}$ .

Solution 5: More Calculus

The probability of this occurring is the volume of the corresponding region within a $1 \times 1 \times 1$ cube, where each point $(x,y,z)$ corresponds to a choice of values for each of $x, y,$ and $z$ . We take a horizontal cross section of the cube, essentially picking a value for z. The area where the triangle inequality will not hold is when $x + y < z$ , which has area $\frac{z^2}{2}$ or when $x+z<y$ or $y+z<x$ , which have an area of $\frac{(1-z)^2}{2}+\frac{(1-z)^2}{2} = (1-z)^2.$ Integrating this expression from 0 to 1 in the form

$\int_0^1 \left(\frac{z^2}{2} + (1-z)^2\right) dz = \bigg[\frac{z^3}{2} - z^2 + z \biggr |_0^1 = \frac{1}{2} -1 + 1 = \frac{1}{2}$

Solution 6: Geometry in 2-D

WLOG assume that $z$ is the largest number and hence the largest side. Then $x,y \leq z$ . We can set up a square that is $z$ by $z$ in the $xy$ plane. We are wanting all the points within this square that satisfy $x+y > z$ . This happens to be a line dividing the square into 2 equal regions. Thus the answer is $\frac{1}{2}$ .

[][] diagram for this problem goes here (z by z square)

Solution 7: More WLOG, Complementary Probability

The triangle inequality simplifies to considering only one case: $\text{the smallest side+ the second smallest side} > \text{the largest side}$ . Consider the complement (the same statement, except with a less than or equal to). Assume (WLOG) $a$ is the largest, so on average $a=1/2$ (now equal to becomes a degenerate case with probability $0$ , so we no longer need to consider it). We now want $b+c<1/2$ , so imagine choosing $b+c$ at once rather than independently. But we know that $b+c$ is between $0$ and $2$ . The complement is thus: $(1/2-0)/2=1/4$ . But keep in mind that we choose each $b$ and $c$ randomly and independently, so if there are $k$ ways to choose $b+c$ together, there are $2k$ ways to choose them separately, and therefore the complement actually doubles to match each case (a good example of this is to restrict b and c to integers such that if $b+c=3$ , then we only count this once, but in reality: we have two cases $1+2$ , and $2+1$ ; similar reasoning also generalizes to non-integral values). The complement is then actually $2(1/4)=1/2$ . Therefore, our desired probability is given by $1-\text{complement}=1/2, C$

Solution 8: 3D geometry

We can draw a 3D space where each coordinate is in the range [0,1]. Drawing the lines $x+y>z, x+z>y,$ and $y+z>x,$ We have a 3D space that consists of two tetrahedrons. One is a regular tetrahedron with side length $\sqrt2$ and the other has 3 sides of length $\sqrt2$ and 3 sides of length $1.$ The volume of this region is $\frac 1 2$ . Hence our solution is $C.$

Video Solution by Punxsutawney Phil

https://youtu.be/DDRRP9VLYO8

Video Solution by The Art of Problem Solving

https://www.youtube.com/watch?v=FqRsTNB89ps&list=PLyhPcpM8aMvI7N78mYZyatqveRU30iNcf&index=3

- AMBRIGGS

@@ Line 67: / Line 67: @@
 We can draw a 3D space where each coordinate is in the range [0,1]. Drawing the lines <math>x+y>z, x+z>y,</math> and <math>y+z>x,</math> We have a 3D space that consists of two tetrahedrons. One is a regular tetrahedron with side length <math>\sqrt2</math> and the other has 3 sides of length <math>\sqrt2</math> and 3 sides of length <math>1.</math> The volume of this region is <math>\frac 1 2</math>. Hence our solution is <math>C.</math>
-=== Solution 9: Cheap Solution ===
-Pretend that the values of <math>x</math>, <math>y</math>, and <math>z</math> are integers ranging from <math>[1,n]</math>. Test out the probability of the first few values of <math>n</math> (for example, <math>P(2) = \frac{5}{8}, P(3) = \frac{15}{27}, P(4) = \frac{34}{64}, P(5) = \frac{65}{125}</math>). Since real numbers contain infinite increments, the answer is the limit as <math>n</math> approaches infinity of <math>P(n)</math>, which is easily hypothesized as <math>\frac{1}{2}</math>, or <math>C.</math>
--solution by fidgetboss_4000 get rect
 == Video Solution by Punxsutawney Phil ==

2016 AMC 12A (Problems • Answer Key • Resources)
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