# Five Tangent Circles

**Discrete Mathematics**Level 4

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 \(\gamma\) is chosen uniformly at random in the interval \([0,2\pi)\). The probability that a half ray drawn from the centre of the original circle at an angle of \(\gamma\) intersects one of the other four circles can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) 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 \(\gamma\) goes only in one direction, not both.

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