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Intermediate Hoop Dreams?

Suppose a basketball player shot below 80% in free-throw percentage in the beginning of the year.

At the end of the year, her average percentage for the year was above 80%.

Does that mean there must have been a moment of time where she had an average of 80% for the year?

NOTE: Underlying this idea is an extension of a "big" math idea. In calculus, we study what is known as the Intermediate Value Theorem for continuous functions. Essentially, the above problem is asking if there is an Intermediate Value Theorem for percentages (and does it have to be specific to free-throws)?


Source: David Ellinger, Seattle Prep (Seattle)...from "apstat listserv"

Hint: Try to prove/disprove it...can you find a counterexample?


Solution Commentary: Comments from David Ellinger, the source of this problem: Start the player (arbitrarily) at 7/10 for the year and start shooting baskets....and prepare to be surprised.

Follow-up Question: What is special about 80% that gives it this property, and what other percentages would share this property?

And can you satisfy your claims with a "Proof"?

In a subsequent e-mail, David added: My intuitive thing says that 80% works because itís of the form (n-1)/n, so its complement is 1/n. So you could prove the complement instead, showing that if you are better than a 1/5 free throw misser, and you become a lower than 1/5 misser, then you will have to pass through 1/5. I assume the algebra would be a little easier.

My comment: Interesting problem....clever commentary!