2016 AMC 12A Problems/Problem 23
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 1: Logic
WLOG the largest number is 1. Then the probability that the other two add up to at least 1 is .
Thus the answer is .
Note: This proof does not explain why WLOG is warranted nor its implications. More detail is necessary
Solution 2: Calculus
When , consider two cases:
1) , then
is the same. Thus the answer is .
Solution 3: 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 4: 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
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