Two points on a circle of radius $1$ are chosen at random. Find the probability that the distance between the two points is at most $1.5.$

kelhaku Jan 18, 2024

#1**0 **

*Two points on a circle of radius $1$ are chosen at random. Find the probability that the distance between the two points is at most $1.5.*

The phrase "the distance between the two points is at most $1**.**5"

can also be stated as "is within 1**.**5." They both mean the same thing.

The first point can be anywhere. Then, we're looking for the probability

that the second point is within 1**.**5 of the first point. I'm assuming that the

distance is measured around the circumference, and not straight across.

The 1**.**5 can be either clockwise or counterclockwise, so there is a 3**.**0

band that the second point can be in.

The circle diameter is 2**.**0 so the entire circumference is 2 • π = 6**.**28.

The probability that the two points are within the 3**.**0 band is 3**.**0 / 6**.**28.

3**.**0 / 6**.**28 = 0**.**48 ... times 100 to make it a percentage is **48%** probability.

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Bosco Jan 18, 2024