+52 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
+743 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.
