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 )
, but without knowledge of , we cannot compute the probability. So, we need to find .
We are given that %3D0.75)
, which translates to %3D0.75)
. 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 %3D0.75)
. 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, %20%3D%20P(-1%3CY%3C1)%20%3D%20P(-1.6651%3C%20Z%20%3C1.6651)%20%3D%20%5CPhi(1.6651)-%5CPhi(-1.6651)%3D.904109)
.
