2016 AMC 12A Problems/Problem 23
Contents
- 1 Problem
- 2 Solution
- 3 Solution 1: Super WLOG
- 4 Solution 2: Conditional Probability
- 5 Solution 3: Calculus
- 6 Solution 4: Geometry
- 7 Solution 5: More Calculus
- 8 Solution 6: Geometry in 2-D
- 9 Solution 7: More WLOG, Complementary Probability
- 10 Solution 8: 3D geometry
- 11 Solution 9(Fastest solution if you have no time): Stick Solution
- 12 Video Solution by Punxsutawney Phil
- 13 Video Solution by The Art of Problem Solving
- 14 See Also
Problem
Three numbers in the interval are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?
Solution
Solution 1: Super WLOG
Because we can let the the sides of the triangle be any variable we want, to make it easier for us when solving, let’s let the side lengths be and . WLOG assume is the largest. Then, , meaning the solution is , as shown in the graph below.
Solution 2: Conditional Probability
WLOG, let the largest of the three numbers drawn be . Then the other two numbers are drawn uniformly and independently from the interval . The probability that their sum is greater than is
Solution 3: Calculus
When , consider two cases:
1) , then
2), then
is the same. Thus the answer is .
Solution 4: Geometry
The probability of this occurring is the volume of the corresponding region within a cube, where each point corresponds to a choice of values for each of and . The region where, WLOG, side is too long, , is a pyramid with a base of area and height , so its volume is . Accounting for the corresponding cases in and multiplies our answer by , so we have excluded a total volume of from the space of possible probabilities. Subtracting this from leaves us with a final answer of .
Solution 5: More Calculus
The probability of this occurring is the volume of the corresponding region within a cube, where each point corresponds to a choice of values for each of and . 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 , which has area or when or , which have an area of Integrating this expression from 0 to 1 in the form
Solution 5.1 (better explanation)
This problem is going to require some geometric probability, so let's dive right in. Take three integers , , and . Then for the triangle inequality to hold, the following inequalities must be true
Now, it would be really easy if these equations only had two variables instead of , because then we could graph it in a -dimensional plane instead of a -dimensional cube. So, we assume is a constant. We will deal with it later.
Now, since we are graphing, we should probably write these equations in terms of so they are in slope-intercept form and are easier to graph.
Now, note that all solutions are in a square in the -plane because , .
I recommend drawing the following figure to get an idea of what is going on. The first line is a line with a negative slope that cuts off a triangle with side length of the bottom left corner of the square. The second line is a line with a positive slope that cuts off a triangle with side length off the top left corner of the square. The third line also has a positive slope and cuts a triangle with side length off the bottom right corner of the square.
Note: All triangles are because the lines have slopes of or .
Using the and signs in the lines, we see that the area that satisfies all three inequalities is the area not enclosed in the three triangles. So, our plan of attack will be to
Find the area of the triangles -> Subtract that from the area of the square -> Use probability to get the answer.
Except, now, we have one problem. is still a variable. But, we want to be a constant. Well, what if we just took the area over every possible value of ? Well, that would be a bit hard, if not impossible to do by hand, but there is a handy math tool that will let us do that: the integral!
To find the area of the triangles, our plan of attack will be
Find the area in terms of -> take the integral from to of the expression for the area (this will cover every possible value of
The area of the triangles is
.
The integral from to is
The total area of all the possible unit squares is quite obviously
Thus, the area not enclosed by the triangles is , and the total area of the square is . Thus, the desired probability is
~Extremelysupercooldude
Solution 6: Geometry in 2-D
WLOG assume that is the largest number and hence the largest side. Then . We can set up a square that is by in the plane. We are wanting all the points within this square that satisfy . This happens to be a line dividing the square into 2 equal regions. Thus the answer is .
[][] 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: . Consider the complement (the same statement, except with a less than or equal to). Assume (WLOG) is the largest, so on average (now equal to becomes a degenerate case with probability , so we no longer need to consider it). We now want , so imagine choosing at once rather than independently. But we know that is between and . The complement is thus: . But keep in mind that we choose each and randomly and independently, so if there are ways to choose together, there are 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 , then we only count this once, but in reality: we have two cases , and ; similar reasoning also generalizes to non-integral values). The complement is then actually . Therefore, our desired probability is given by
Solution 8: 3D geometry
We can draw a 3D space where each coordinate is in the range [0,1]. Drawing the lines and We have a 3D space that consists of two tetrahedrons. One is a regular tetrahedron with side length and the other has 3 sides of length and 3 sides of length The volume of this region is . Hence our solution is
Solution 9(Fastest solution if you have no time): Stick Solution
Consider a stick of length . Cutting the stick at two random points gives a triangle from the three new segments. These two random points must be on opposite sides of the halfway mark. Thus, after the first cut is made, there is probability that the second cut is on the opposite side.
-this isn't necessarily true: what if the cuts are very very close to the edges? -happypi31415
-alanisawesome2018
Video Solution by Punxsutawney Phil
Video Solution by The Art of Problem Solving
https://www.youtube.com/watch?v=FqRsTNB89ps&list=PLyhPcpM8aMvI7N78mYZyatqveRU30iNcf&index=3
- AMBRIGGS
See Also
2016 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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