This is my solution to the riddle “Should You Shoot Free Throws Underhand?” I apologize for the ugly mathematical mark-up.
We see , and where and are normal with and unknown.
The question asks for <img alt="P(-1<Y, but without knowledge of , we cannot compute the probability. So, we need to find .
We are given that , which translates to . If we divide both sides of the inequality inside the probability by , then we have the sum of two standard normal random variables squared, which is a chi-square random variable with 2 degrees of freedom (you can find a reference for that here).
Thus, with , we have . Using some tables or technology, we can find the 75th percentile of a chi-square distribution with 2 df to be 2.772589, which when set equal to , will reveal that .
Now, <img alt="P(|Y|<1) = P(-1<Y<1) = P(-1.6651< Z .