Breaking a Stick

Probability Level 3

I have a stick of a certain length. I use my hands to break the stick into 2.

The probability that at least 1 of the 2 smaller sticks is at least half the length of the original stick is:

0% 25% 50% 100% 75%

Winston Choo by Winston Choo , 46, Singapore

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1 solution

Jordan Cahn
Dec 13, 2018

Let the length of the original stick be L L , with shorter sticks of length l 1 l_1 and l 2 l_2 . If l 1 < 1 2 L l_1<\frac{1}{2}L and l 2 < 1 2 L l_2<\frac{1}{2}L then l 1 + l 2 < 1 2 L + 1 2 L = L l_1+l_2 < \frac{1}{2}L + \frac{1}{2}L = L But l 1 + l 2 = L l_1+l_2 = L , a contradiction. Therefore either l 1 ≥ 1 2 L l_1\geq \frac{1}{2}L or l 2 ≥ 1 2 L l_2\geq \frac{1}{2}L and the desired probability is 100 % \boxed{100\%} .

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