The numbers $x_1,$ $x_2,$ $x_3,$ $x_4$ are chosen at random in the interval $[0,1].$ Let $I$ be the interval between $x_1$ and $x_2,$ and let $J$ be the interval between $x_3$ and $x_4.$ Find the probability that intervals $I$ and $J$ both have length greater than $3/4$.

sandwich Oct 9, 2023

#1**0 **

To find the probability that intervals I and J both have length greater than 3/4, we can use the following steps:

Calculate the probability that interval I has length greater than 3/4. Since x1 and x2 are chosen independently at random in the interval [0,1], the length of interval I is uniformly distributed between 0 and 1. Therefore, the probability that interval I has length greater than 3/4 is 1−01−3/4=1/4.

Calculate the probability that interval J has length greater than 3/4. Using the same logic as in step 1, we can calculate that the probability that interval J has length greater than 3/4 is also 1/4.

Multiply the two probabilities to find the probability that both intervals I and J have length greater than 3/4. Since x1, x2, x3, and x4 are chosen independently at random, the events that interval I has length greater than 3/4 and interval J has length greater than 3/4 are independent. Therefore, the probability that both intervals I and J have length greater than 3/4 is the product of the two probabilities we calculated in steps 1 and 2, which is 1/4 * 1/4 = 1/16.

bingboy Oct 9, 2023

#2**+1 **

.As Melody often notes, it can be useful to plot these sort of probabilities against each other. Consider x_1 and x_2:

The probabilities of the pair can be thought of as uniformly randomly scattered points within a unit square.

The only regions of the square in which the lengths between the two values are greater than 3/4, are within the two red triangles indicated in the image above.

The probability that the lengths are greater than 3/4 is therefore given by p = Area of triangles/Area of square

The same probability holds for the lengths between x_3 and x_4, so, because the two probabilities are independent, the overall probability is given by p^2.

I'll leave you to do the number crunching.

Alan Oct 11, 2023