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 ã is chosen uniformly at random in the interval [0,2ð). The probability that a half ray drawn from the centre of the original circle at an angle of ã intersects one of the other four circles can be expressed as ab, where a and b are coprime positive integers. What is the value of a+b?

To solve this problem, we need to determine the probability that a half ray drawn from the center of the original circle intersects one of the four surrounding circles.

Let's visualize the scenario:

- Start with the original circle with radius 1.
- Draw four non-overlapping circles externally, each with radius 1, that are tangential to the original circle. These four circles are placed at the top, right, bottom, and left of the original circle, forming a square arrangement.

Now, let's analyze how the half ray drawn from the center of the original circle can intersect one of the four surrounding circles:

- For the half ray to intersect one of the surrounding circles, it must fall within one of four regions between adjacent circles.
- Each region comprises two arcs: the arc on the outer circle and the arc on the inner circle (between adjacent circles).
- In total, there are eight arcs separating the regions between the surrounding circles.
- The length of each arc is given by the formula: θ = radius / circle's diameter. In this case, θ = 1/2.

Now, let's reason about the probability of the half ray intersecting one of the surrounding circles:

- The angle ã, uniformly chosen in the interval [0, 2π), represents the random selection of the direction of the half ray.
- To compute the probability, we need to determine the fraction of angles ã that cause the half ray to intersect one of the surrounding circles.
- This fraction is given by the ratio of the total arc length of the regions between the surrounding circles to the total circumference of the original circle. The circumference is 2πr = 2π(1) = 2π.

The total arc length of the regions between the surrounding circles is given by:

8 * θ = 8 * (1/2) = 4π.

Therefore, the probability of the half ray intersecting one of the surrounding circles is:

Probability = (arc length of the regions) / (circumference of the original circle) = (4π) / (2π) = 2.

Now we have the probability expressed as 2/1, which can be written as 2.

Thus, the values of a and b in the fraction a/b are 2 and 1, respectively.

The sum of a and b is 2 + 1 = 3.

Therefore, the value of a + b is 3.