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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 $3/4$.

 Jan 31, 2024
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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, \(\fbox{1/4}\)

 Jan 31, 2024

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