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?
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galat answer hai ye gadhe
wrong
wrong
136, is it correct???
171
To find the expected number of pairs of chords that intersect, we can use linearity of expectation.
Let's consider any two chords C_i and C_j. For them to intersect, their endpoints must be placed in a specific order around the unit circle.
Without loss of generality, let's fix chord C_i and consider the possible positions for one of its endpoints. We can place it anywhere on the unit circle, which gives us 1 position.
Now, let's consider the second endpoint of C_i. It must be placed in one of the remaining 2 units on the circle (since one unit is occupied by the first endpoint and the other units are occupied by the remaining chords' endpoints).
Similarly, the first endpoint of C_j can be placed in 1 position, and the second endpoint in 2 positions.
Therefore, the probability that C_i and C_j intersect is (1/2) * (1/2) = 1/4.
Now, let's calculate the expected number of pairs of chords that intersect.
We have 18 chords, so there are a total of C(18, 2) = 153 pairs of chords.
The probability that any pair of chords intersects is 1/4.
Therefore, the expected number of pairs of chords that intersect is:
153 * (1/4) = 38.25
Therefore, the expected number of pairs of chords that intersect is 38.25.