March 26, 2021
The rules for men’s basketball in the Riddler Collegiate Athletic Association’s (RCAA) are a little different from those in the NCAA. In the NCAA, when a player is fouled attempting a 3-point shot and misses, they always get three free throw attempts, regardless of how many fouls the opposing team has committed.
But in the RCAA, a player must earn each additional foul shot by making the previous one (similar to the “bonus” rules of the NCAA mentioned in the Express). In other words, a player can take a second shot if they make the first, and they can take a third shot if they make the second.
Suppose a player on your team has a known shooting profile: Their probability of making the first free throw is
What is the greatest number of distinct shooting profiles that are made up of these three different probabilities —
Answer: Two profiles.
Explanation:
Let
Now, there are exactly
Lemma: two profiles cannot produce the same expected score if both of them contain at least one probability in the same position.
Proof: Clearly if two of the three probabilities are in the same position, the third one must be as well. Therefore, without loss of generality, let
Corollary: this means that at most
Now let's formally prove that two profiles is the greatest number of distinct shooting profiles.
Let's consider two profiles
So we can pick
Let's verify those profiles yield the same expected score:
Hence achievability part is proven!
Now we have to show that the two profiles answer is the tightest upper bound. To do this, we show that it is not possible to have three distinct profiles, as by our lemma we showed it is not possible to have more than three. Assuming, by contradiction, that three distinct profiles is possible, then by using our lemma from earliear this can only result from:
Re-arranging for
We now substitute the above equation into
Since it is given that
Therefore, two distinct profiles is the tightest upper bound. Q.E.D.