Two points are chosen uniformly at random on the unit circle and joined to make a chord C1. This process is repeated 3 more times to get chords C2,C3,C4. The probability that no pair of chords intersect can be expressed as a/b where a and b are coprime positive integers. What is the value of a+b?

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171 To solve this problem, we need to find the probability that no pair of chords intersect. Let's break down the problem step by step.

Step 1: Choosing points on the unit circle:
We have two points chosen uniformly at random on the unit circle. The probability of choosing any specific point on the circle is equal since points are chosen uniformly. Thus, the probability of choosing any point on the unit circle is 1.

Step 2: Joining points to form chords:
With the two points, we join them to form a chord. Each chord has equal probability of being formed since any combination of two points on the circle is equally likely to be chosen. So, the probability of forming each chord is 1/2.

Step 3: Repeating the process three more times:
We repeat the process three more times to form a total of four chords. The probability of forming each chord is still 1/2, and since each step is independent, the probability for all four chords is obtained by multiplying the individual probabilities:

Probability of forming all four chords = (1/2) * (1/2) * (1/2) * (1/2) = 1/16

Step 4: Checking for intersections:
Now, we need to consider the condition that no pair of chords intersect. To achieve this, we can imagine the unit circle divided into four regions created by the four chords (C1, C2, C3, and C4). In order for no pair of chords to intersect, each of these regions should be empty.

If the chords do not intersect, they form a cyclic quadrilateral inside the unit circle. The number of regions created by the chords is related to the number of intersection points inside the unit circle, which is given by the number of regions minus 1.

In this case, we have four chords (C1, C2, C3, and C4), which create five regions:

Number of regions = 5

So, the number of intersection points is:

Number of intersections = Number of regions - 1 = 5 - 1 = 4

Step 5: Determining the probability of no intersections:
Since we have 4 intersection points and 16 possible ways the chords can be formed (each chord has 2 possibilities), we can determine the probability of having no intersections.

Probability of no intersections = Number of ways to form chords with no intersection / Total number of possible ways to form chords

We need to find the number of ways to form chords with no intersection. This is equivalent to counting the number of ways to arrange 4 intersection points among 16 possible positions on the chords.

Using combinatorics, this can be calculated as:

Number of ways to arrange 4 intersection points among 16 possible positions = 16! / (4! * (16 - 4)!) = 16! / (4! * 12!)

To find the total number of possible ways to form chords, we have 2 possibilities for each chord, repeated 4 times:

Total number of possible ways to form chords = 2^4 = 16

Thus, the probability of no pair of chords intersecting is:

Probability of no intersections = (16! / (4! * 12!)) / 16 = 1/495

Finally, we have a = 1 and b = 495. Therefore, a + b = 1 + 495 = 496.