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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 x1,x2,x3,x4 are chosen at random in the interval [0,1]  Let I be the interval between x1,x2 and let J be the interval between x3,x4  Find the probability that intervals I,J both have length greater than 34

 

Ok, so this implies that x2x1>3/4, same thing for the other pair.

 

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

This means that x1<1/4, which is 1/4 chance, times the chance that x2>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 x3,x4 and get the same result. Adding these together will give us our final answer, 1/4

 Jan 31, 2024

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