Suppose you break a stick of unit length at a point chosen uniformly at…
2014
Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .
Attempted by 26 students.
Show answer & explanation
Correct answer: 0.24 to 0.27
Key idea: Let X be the break point, uniformly distributed on [0,1]. The length of the shorter piece is S = min(X, 1 - X).
Write the expectation as an integral: E[S] = ∫_0^1 min(x, 1 - x) dx.
Split the integral at 1/2, since for x in [0,1/2] the minimum is x, and for x in [1/2,1] the minimum is 1 - x.
Compute the two integrals: ∫_0^{1/2} x dx = [x^2/2]_0^{1/2} = 1/8, and ∫_{1/2}^1 (1 - x) dx = 1/8.
Sum them to get E[S] = 1/8 + 1/8 = 1/4 = 0.25.
Conclusion: The expected length of the shorter stick is 1/4 = 0.25.