re: Shooting free throws underhand is superior; However almost no one does it

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The percentage of time you’ll make a “granny-style” free throw is about 90.4 percent. Instead of fixing the radius of the circle, I fixed the variance of the random variables to one. So I have to find the radius of the circle first. The circle equation, thinking back to high school geometry, is x2+y2=r2. The distribution of x2+y2 is a Chi-squared distribution with 2 degrees of freedom. Thus, to find the radius, I solved the following equation for r: 0.75=1-e-r2/2. That makes r˜1.665. Using this value, I can find p=F(1.665)-F(-1.665), where F(x) is the cumulative distribution function of a standard normal variable. This gives the solution of 0.904.

In other words, if you’re a 75-percent free-throw shooter using the “normal” method, you can expect to improve to above 90 percent shooting underhand!

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