# Probability the longer segment is at least twice as long as the shorter

A point is selected at random from the interval (0; 1); it then divides this interval into two segments. What is the probability that the longer segment is at least twice as long as the shorter segment?

Do we treat it as a uniform random variable and use the normalization rule to get f(x) and F(x)? I'm not really sure.

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Could you give a title which indicates what the question is about? –  Asaf Karagila Sep 19 '12 at 13:07

## 1 Answer

Think: which points will divide the interval into two segments, one at least twice as long as the other? and, what is the probability then of choosing one of those points?

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@ Gerry Myerson-I suppose the points between 0 and 1/3 and then 2/3 and 1 would split it into two with one at least double the length of the other. If this is the case the longer side would be at least 2/3 of the interval. Does this mean there is a 2/3 chance? –  Sprock Sep 19 '12 at 18:41
The points between 0 and 1/3, together with the points between 2/3 and 1, constitute two-thirds of the interval $(0,1)$; that means there's a 2 out of 3 chance. –  Gerry Myerson Sep 20 '12 at 1:06