A stick of unit length is broken at random. What is the probability that a longe

Question

A stick of unit length is broken at random. What is the probability that a longer piece is more than twice a length of the shorter piece: confirm it by simulations in R, and demonstrate that the distribution of the long and short piece is uniform by plotting the empirical cdfs on one graph side-by-side using par(mfrow=c(1,2)) command. Prove that the distributions are uniform. [Hint: Use the fact that max(X, 1-X) <= y is equivalent to X <= y and 1-X <= y, and elementary probability formula Pr(A Uinon B) = 1 – Pr(A Intersection B) when computing the cdf for the shortest piece.]

Explanation / Answer

Solution:

. We shall model the stick as as interval [0, L], where L is the length of the stick in some convenient unit, say meters.

Then we will model breaking the stick as randomly choosing a point X in the interval.

The longer piece of the stick will be at least twice as long as the shorter piece precisely when either 0 X L/3 or 2L/3 L.

The meaning of choosing the breaking breaking point “at random” is at issue.

We cannot capture the idea that each point of the interval is equally to be chosen by assigning the same positive probability to each point, for the sum of these probabilities would not even be finite.

Instead of thinking about the probability of choosing an individual point of the interval, we focus on defining the probability that the point chosen lies in a given subinterval of the interval [0, L].

If [a, b] is a subinterval of [0, L] we will say the probability that X is in [a, b] is proportional to the length of [a, b].

That is, this probability is given by (b a)/L .

This definition captures the intuitive idea that the point chosen is equally likely to be anywhere in the interval.

Since the event that the longer piece of the stick is at least twice as long as the shorter piece is {0 X L/3} {2L/3 X L},

we see that the required probability is 2/3.

The probability that a longer piece is more than twice a length of the shorter piece is 2/3

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