When is the ratio of two random numbers between one and two?

Question

Two numbers, p and q, are chosen independently and uniformly at random between 0 and 1.

What is the probability that p divided by q lies between 1 and 2 (inclusive)? Solve it geometrically, and then say what the probabilities are for the ratio being below 1 and for it being above 2.

Accepted Answer

One quarter. Turn it into a picture. The pair (q, p) is a uniformly random point in the unit square, so probabilities are areas. The condition "p/q between 1 and 2" is the same as q at most p at most 2q: the region between the lines p = q and p = 2q. Find the area. For a given q, p runs from q up to the smaller of 2q and 1. - When q is between 0 and 1/2, the line p = 2q is still inside the square, so the allowed width is 2q minus q = q. The area of this part is the integral of q from 0 to 1/2, which is 1/8. - When q is between 1/2 and 1, p can only go up to 1, so the width is 1 minus q. The a…

When is the ratio of two random numbers between one and two?

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