Question

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?

Answer 1

137

Answer 2

349

Answer 3

How

Answer 4

both r incorrect

Answer 5

51

Answer 6

[AI-generated answer — Explain Bot]

To find the expected number of pairs of chords that intersect, we can consider each pair of chords and calculate the probability that they intersect. Then, we can sum up these probabilities to find the overall expected number.

Let's start by considering a single pair of chords. Any pair of chords will intersect if and only if the endpoints of one chord lie on opposite sides of the other chord. There are two possible cases to consider: (1) the two chords cross each other, or (2) one chord is completely inside the other.

Case 1: Crossed chords
In this case, the probability that the endpoints of one chord lie on opposite sides of the other chord is 1/2. This is because there are two possible ways for the endpoints to be placed such that they are on opposite sides. Regardless of the positions of the first two chords, the probability remains constant.

Case 2: Chords with one inside the other
In this case, the probability that the endpoints of one chord lie on opposite sides of the other chord is 1. This is because one chord is entirely enclosed by the other chord, so the endpoints will always be on opposite sides.

Now, let's consider the total expected number of pairs of chords that intersect. Since each pair of chords can be either crossed or one inside the other, we need to add up the expected number of pairs for each case.

Case 1: Crossed chords
For the first pair of chords, there are 18 possible chords to choose from. If we fix one chord, there are 17 chords remaining. Therefore, the probability that any given pair of chords is crossed is 18/34. So, the expected number of pairs of crossed chords can be calculated as:

18/34 * 17 = 9

Case 2: Chords with one inside the other
For the first pair of chords, there are 18 possible chords to choose from. If we fix one chord, there are 17 chords remaining. Therefore, the probability that any given pair of chords has one inside the other is 17/34. So, the expected number of pairs of chords with one inside the other can be calculated as:

17/34 * 17 = 8.5

Now, we can sum up the expected number of pairs for each case:

9 + 8.5 = 17.5

Therefore, the expected number of pairs of chords that intersect is 17.5.