This blog post is the closest I will ever get to playing basketball and it’s still pretty far away.
According to the internet, free throws are awarded to a player fouled in the act of shooting and when the defending team has committed 5 team fouls.
It then makes sense that there is a thing called a free throw percentage (FTP). Basically, for a given player, it’s the percentage of free throws they get that end up going in the ring. The best NBA players often have percentages above 90%.
Here’s something for you to think about:
Early in a season, a player has a FTP of below 90%. Later in the season, the same player has a FTP of above 90%. Is there a point in the season when their FTP is exactly 90%?
When I was asked this, my answer was: maybe, maybe not. The Intermediate Value Theorem sprung to mind:
A continuous function on must take on any given value between and at some point within the interval.
Of course, a player’s FTP is not continuous, so the IVT does not apply here.
For example, if they have scored 5 out of 8 free throws, then their FTP is equal to 5/8 or 62.5%. Say they score their next free throw; then their FTP becomes 6/9 or roughly 66.67%.
Therefore, surely it’s possible for a FTP to jump over 90% without equalling it.
It turns out, however, that 90% is unavoidable. That is, the player in question will have a FTP of exactly 90% at some point.
Proof. Let’s prove this with a little proof by contradiction. We start by assuming that they can jump over 90%. That is, one second they are below 90% and the next they are above it.
We will denote by the number of free throws they have been awarded when they are sitting just below 90% and we will denote by the number of these free throws that they have scored.
Thus, their current FTP is equal to .
For this to be above 90% on their next free throw, they must score on the next free throw. Their FTP now becomes . By our assumption, this is above 90%. This gives us the inequality
We can rearrage the leftmost inequality to get .
We can rearrange the rightmost inequality to get .
Then, putting these together gives us the inequality
However, this is a contradiction, for , and are all integers. Furthermore, and are consecutive integers, and so there is no way you can squeeze another integer between them!
Therefore, the player’s FTP must be exactly equal to 90% at some point.
This is pretty wild stuff. It blew my noodle when I saw it.
Also, is not that special! The result works for any number of the form where is an integer. And being the avid blog reader that you are, I’ll end this blog here so that you can prove it.