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# counting

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The numbers x1, x2, x3, and x4 are chosen at random in the interval [0, 1] Let I be the interval between x1 and x2 and let J be the interval between x3 and x4 Find the probability that intervals I and J overlap.

it is not 1/2 or 1/12

Jul 14, 2024

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Understanding the Problem

We have four random points on the unit interval [0, 1]. We form two intervals, I and J, from these points. We want to find the probability that these intervals overlap.

Approach

To calculate the probability, we'll consider the complementary event: the probability that the intervals do not overlap. If we can find this probability, we can simply subtract it from 1 to get the desired probability.

Calculating the Probability of Non-Overlapping Intervals

For the intervals to not overlap, one interval must completely lie to the left of the other. There are two possibilities for this:

I is entirely to the left of J:

x1 < x2 < x3 < x4

Probability of this arrangement is 1/4! (since there are 4! equally likely orderings of the four points).

J is entirely to the left of I:

x3 < x4 < x1 < x2

Probability of this arrangement is also 1/4!

The total probability of non-overlapping intervals is the sum of these two probabilities:

P(non-overlapping) = 1/4! + 1/4! = 2/4! = 1/12

Calculating the Probability of Overlapping Intervals

Finally, the probability of overlapping intervals is:

P(overlapping) = 1 - P(non-overlapping) = 1 - 1/12 = 11/12

Therefore, the probability that intervals I and J overlap is 11/12.

Jul 14, 2024