I read somewhere this is a Google interview question:
Let’s say you drop a stick, breaking it randomly in 2 places, leaving you with 3 smaller sticks. What’s the probability you can make a triangle out of the 3 resulting sticks?
There are a number of ways to tackle this, including this very elegant solution: http://www.cut-the-knot.org/Curriculum/Probability/TriProbability.shtml. For the less elegantly inclined (and those as unlikely as me to be working at Google any time soon), here’s another way:
The first insight is that we can think of the breaks as two separate events. The first break can happen anywhere. To make a triangle, the second break has to happen on the opposite half from the first break. Otherwise we’d end up with one stick being more than half the original length, and we can’t make a triangle if that’s the case. The probability of the 2nd break being on the opposite half is 50%.
But even with this first condition met, we may still end up with one stick being longer than half the original, if the two breaks are close to the ends. We also need the sum of the distances of the two breaks from the ends to be greater than 1/2 the length of the original stick. What’s the probability of that? The probability that two random numbers from 0 to .5 add up to more than .5 is 50%
So that’s a slightly long-winded way of arriving at the answer – the 2nd break has to satisfy two independent 50% chances, so we have a 1 in 4 chance of being able to make a triangle.
Google, you have my number.