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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 $1.5.$

 Jan 18, 2024
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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.  

.

 Jan 18, 2024

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