Two points are chosen uniformly at random on the unit circle and joined to make a chord C1. This process is repeated 17 more times to get chords C2,C3,…,C18. What is the expected number of pairs of chords that intersect?
To find the expected number of pairs of chords that intersect, we need to determine the probability that each pair of chords intersects and then sum up these probabilities.
Let's consider the first chord, C1. It doesn't intersect with any other chord initially. Now, when we add the second chord, C2, the probability that C1 and C2 intersect can be visualized as dividing the unit circle into 4 equal arcs. C2 must intersect with C1 if it falls in the two arcs that lie between the endpoints of C1. Hence, the probability that C1 and C2 intersect is 2/4.
Similarly, for each subsequent chord added, the probability that it intersects with any of the previous chords can be visualized as dividing the unit circle into n+2 equal arcs, where n is the number of chords already added. The chord must intersect with any of the n previous chords if it falls in the n arcs that lie between the endpoints of each chord. Therefore, the probability that any new chord intersects with any of the previous chords is n/(n+2).
To find the expected number of pairs of chords that intersect, we sum up these probabilities for all chords from C2 to C18:
E(intersect) = (2/4) + (3/5) + (4/6) + ... + (17/19)
To simplify this expression, we can rewrite it as:
E(intersect) = 2(1/4 + 1/5 + 1/6 + ... + 1/19) - 1
Now, we can use the fact that the harmonic series 1/1 + 1/2 + 1/3 + ... + 1/n grows logarithmically to approximate the sum 1/4 + 1/5 + 1/6 + ... + 1/19.
Since the given series goes from 1/4 to 1/19, we have a total of 16 terms.
Approximately, the harmonic series can be approximated as:
1/1 + 1/2 + 1/3 + ... + 1/n ≈ ln(n) + γ, where γ is the Euler-Mascheroni constant.
Therefore, the expected number of pairs of chords that intersect is approximately:
E(intersect) ≈ 2(ln(19) + γ) - 1
Using a calculator, we can find the approximation for E(intersect) to be approximately 3.410.
Hence, the expected number of pairs of chords that intersect is approximately 3.410.