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 $3/4$.

booboo44 Jan 31, 2024

#1**0 **

**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 \(x_1,x_2\) and let \(J\) be the interval between \(x_3,x_4\) Find the probability that intervals \(I,J\) both have length greater than \(\frac{3}{4}\). **

**Ok, so this implies that \(x_2-x_1>3/4\), same thing for the other pair.**

**We'll just assume that the numbers go from least to greatest.**

**This means that \(x_1<1/4\), which is 1/4 chance, times the chance that \(x_2>3/4\), which is also 1/4. multiplying, we get 1/16, which we multiplu by 2 since they don't have to go from least to greatest, we just assumed. This gives us 1/8.**

**We do the same for \(x_3,x_4\) and get the same result. Adding these together will give us our final answer, \(\fbox{1/4}\)**

Imcool Jan 31, 2024