The Mendoza Line

Question

At some point in his career, Mario Mendoza's batting average was above 0.200. Sometime later, it was below 0.200. Prove that, at some point, he was batting exactly 0.200.

For those unfamiliar with baseball: players have a series of "at-bats" (attempted hits) which either succeed, or don't. At any point, a player's batting average is the fraction of their at-bats so far which were hits. This number is usually rounded to 3 decimals, but for the purposes of this puzzle, consider all fractions to be exact.

For those familiar with baseball: the point where he is batting 0.200 might occur during a game, so assume that a player's batting average is updated after every at-bat.

Unnecessary fun fact: Mario Mendoza was a real player, whose batting average tended to be around 0.200, an embarrassingly low rate. This is pretty led his teammates to joke that anyone who batted below 0.200 was "below the Mendoza line."

Answer 1

Let $h$ be the number of successful hits, $t$ the number of tries. Initially $\frac{h}{t} > \frac{1}{5}$ or equivalently, $5h > t$. At any point we may simultaneously add 1 to $t$ and add 1 or 0 to $h$. Notice, if $5h > t$, then $5(h+1) > t+1$ certainly, and if $5h > t$, then $5(h+0) \ge t+1$ certainly (as we are working with integers). Thus to pass from $5h > t$ to $5h < t$, we must pass through $5h = t$.

This argument is valid for any $\frac{1}{n}$, but not any $\frac{a}{b}$, as then we cannot argue that if $bh > at$ then $b(h+0) \ge a(t+1)$.

Answer 2

All this theoretical stuff is nice, but why would we do that when we can check actual data and determine whether the statement is true or false?

All stats come from www.baseball-reference.com.

Looking at Mario Mendoza's career numbers shows that his first season, he hit .221. This shows that at one point in his career, his batting average was above .200. The next 3 seasons, he hit below .200 each season. However, end of season totals keep his career batting average above .200. Therefore, it is necessary to look at his batting average within the seasons to determine whether the statement is true.

After his 3rd season (1976), he had 305 at bats and 62 hits, for a career batting average of .203. In 1977, after 10 at bats, he had 1 hit. This gives him 63 hits in 315 at bats, for a career batting average of .200. This proves the second part, that at one point he was batting exactly .200.

(Un)fortunately, his next at bat was a hit, keeping him over .200. Continuing into the season, we see that over the next 10 at bats (including that hit), he had 2 hits, giving him 65 hits in 325 at bats, back to exactly .200.

Then on July 9th, 1977, he entered the game as a pinch runner in the 8th inning. In the 10th inning, he grounded out. This made him 65 for 326 in his career, for a batting average of .199. This proves that after he was hitting .200, he was hitting below .200.

In the bottom of the 12th inning, Mendoza hit a walk-off single to win the game for the Pirates (9-8 over the Phillies). This would raise his batting average to .202. He then managed to keep his career batting average over .200 for the rest of the season.