Problems in Probability

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Fall 2011 Homework 7: Question 1
Consider the following:
(a) A stick is broken into three pieces by picking two points independently and uniformly along the stick, and breaking the stick at those two points. What is the probability that the three pieces can be assembled into a triangle? Hint: a triangle can be formed from 3 line segments of lengths a, b, c if and only if a, b, c (0, 1/2). The probability can be interpreted geometrically as proportional to an area in the plane, avoiding all calculus, but make sure for that approach that the distribution of the random point in the plane is Uniform over some region.
(b) Three legs are positioned uniformly and independently on the perimeter of a round table. What is the probability that the table will stand?
Solution: (a) 1/4. Note that the idea of interpreting probabilities as areas works here because (X, Y) is Uniform on the square. For other distributions, in general we would need to find the joint PDF of X, Y and integrate over the appropriate region. (b) Think of the legs as points on a circle, chosen randomly one at a time, and choose units so that the circumference of the circle is 1. Let A,B,C be the arc lengths from one point to the next (clockwise, starting with the first point chosen). Then P(table falls) = P(the 3 legs are all contained in some semicircle) = P(at least one of A, B, C is greater than 1/2) = 3/4, by Part (a). So the probability that the table will stand is 1/4. (Can also obtain this answer by alternatively, letting Cj be the clockwise semicircle starting from the jth of the 3 points. Let Aj be the event that Cj contains all 3 points. Then P(Aj) = 1/4 and with probability 1, at most one Aj occurs.) See iTunes course for full solution.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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