A circle of radius 1 is drawn in the plane. Four non-overlapping circles each of radius 1, are drawn (externally) tangential to the original circle. An angle γis chosen uniformly at random in the interval [0,2π). The probability that a half ray drawn from the centre of the original circle at an angle of γ intersects one of the other four circles can be expressed as a/b, where aand b are coprime positive integers. What is the value of a+b?

Question

A circle of radius 1 is drawn in the plane. Four non-overlapping circles each of radius 1, are drawn (externally) tangential to the original circle. An angle γis chosen uniformly at random in the interval [0,2π). The probability that a half ray drawn from the centre of the original circle at an angle of γ intersects one of the other four circles can be expressed as a/b, where aand b are coprime positive integers. What is the value of a+b?

Details and assumptions

The half ray from the centre of the fifth circle at angle γ goes only in one direction, not both.

Steve · Answer

Well, a unit circle externally tangent to the central unit circle subtends a central angle of 60°.

Draw a diagram to see that the tangents from the center to the external circles form angles of 30° each.

So, if there are 4 external circles, they cover 240°.

Since there are 360° available, the chance that a central ray will intersect one of the external circles must then be 240/360 = 2/3