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# Hard probability

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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