Intersecting Intervals

Probability Level 3

Choose 2 numbers uniformly from $[0,1]$ and let them be the endpoints of the first interval.
Choose another 2 numbers uniformly from $[0,1]$ and let them be the endpoints of the second interval.

What is the probability that these two intervals intersect?

\frac{2}{3}

\frac{1}{2}

\frac{1}{3}

\frac{3}{4}

1 solution

Chung Kevin
Sep 30, 2016

Here is a fast way of seeing it, instead of trying to condition on "choose 2 numbers uniformly at random".

Assume that the 4 endpoints are fixed, and we have $a < b < c < d$ . Consider the possible intervals:

(a,b),(c,d)
(a,c),(b,d)
(a,d),(b,c)

2 out of 3 of these possible intervals will result in an intersection.

Since this is true for all combination of end-points, this is also true of the total probability.

Intersecting Intervals

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