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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.

 May 2, 2023
 #1
avatar+135 
+1

The probability that the distance between two points on a circle of radius 1 is at most 1.5 is approximately 0.477 or 47.7%, which corresponds to the range of central angles less than or equal to 3 radians out of the total central angle of 2π radians.

Step-by-step explanation:

Let's consider two points on a circle of radius 1, labeled A and B. The distance between A and B is given by the length of the arc AB on the circle, which is a fraction of the circumference of the circle. The circumference of a circle of radius 1 is 2π, so the length of the arc AB is given by

d = θ/2π * 2πr = θ

where d is the distance between A and B, θ is the central angle between A and B (in radians), and r is the radius of the circle (which is 1 in this case).

To find the probability that the distance between A and B is at most 1.5, we need to find the range of central angles that correspond to arc lengths less than or equal to 1.5. Let's call this range of angles θ₁. We have

θ₁ = 2d/r = 2(1.5)/1 = 3

This means that the central angle between A and B must be less than or equal to 3 radians for the distance between A and B to be at most 1.5. Since the total central angle of the circle is 2π radians, the probability of selecting two points with a central angle less than or equal to 3 radians is

P(θ ≤ 3) = θ/2π = 3/2π ≈ 0.477

Therefore, the probability that the distance between two points on a circle of radius 1 is at most 1.5 is approximately 0.477 or 47.7%.

 May 2, 2023
 #2
avatar+118616 
+1

The probability that the distance will be MORE than 1.5cm   will be 

the area of the solid red section / the area of the unit circle.

The prob that it will be LESS than 1.5cm will be 1 - the answer you just got.

 

The answer will be way higher than the  47%  acyclics figured.

 

If you want me to explain further how I figure this then ask me in a continuing post.

 

 

 

 May 12, 2023

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