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Given $x,y$ are uniformly selected at random in the interval $(0,1),$ compute the probability that $y>x^2.$

This is routine with integration, but the answer seems so simple that I can't help but wonder if there's a more "elementary" method (i.e. one that does not require calculus.)

Does such a method exist?

David G. Stork
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x3yukari
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    I suppose any method of calculating the area bounded by a parabola works, and calculus just so happens to be the ideal tool for this. But something like https://math.stackexchange.com/q/305268, sure, appealing to Archimedes. – Jakob Streipel Nov 27 '21 at 19:17

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One calculus-free method I would recommend always as it is a check on your assumptions (and math) is do a simple Monte Carlo simulation, for example, in a spreadsheet setting.

Here one could simply generate a pair of random uniform deviates via the worksheet provided RAND() function that creates a Uniform random deviate between 0 and 1.

Next, add a logic function to a cell, for example, as we are interested in the probability that $y > x^2$, one such formula would be $=if(RAND()>(RAND())^2,1,0)$. Copy this formula to a block say of a thousand cells.

Finally, a new result cell containing simply the SUM(range) formula divided by a COUNT(range) to arrive at the first simulation estimated probability value. Copy this number (as a value) to a new data column (best performed by writing worksheet macro for the advanced student).

As required no calculus, but this is just a purely based numerical answer which can suffer from sampling statistical error. If you know the theoretical value, see if further simulations indicate convergence, otherwise, based on a statistical t-test, for example, it may be clear that the theoretical based solution is off, or check your worksheet for any possible errors.

I trust this helps if only in providing the answer.

AJKOER
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