Mike Trout Numerology

Question: in one of the great numerological miracles of our time, baseball players are obtaining batting averages with the same first three significant digits as the number of games in the stretch that the batting statistics are taken over. Is this an effect of rare planetary alignment? Suppose that a baseball player gets $4$ at bats in every of the $N$ games they play. What is the greatest value of $N$ for which it is impossible for the digits of their batting average (rounded to the nearest thousandth) to equal the number of games they’ve played in.

(FiveThirtyEight)

Solution

Batting average is the probability that a batter gets a hit in any given at bat, i.e.

If the player has $h$ hits on the $N$ game season, and therefore $4N$ at bats, then their batting average is

The numerological phenomena we’re interested in is the case when their batting average, rounded to the nearest thousandth, has the same digits as the number of games played. In other words, if the batting average is $x$ then we want to know if

If a number rounds to $N,$ it means that it’s either up to $1/2$ less than it or up to just less than $1/2$ greater than it:

Multiplying through to isolate $h,$ this becomes

This is the condition on $h$ that must be satisfied for the numerological miracle to occur. However, $h$ also has to be an integer, which means that the left and right hand sides of the inequality have to be on opposite sides of an integer. The gap between these two sides is $N/250,$ which is greater than $1$ so long as $N$ is greater than $250.$

The phenomena can fail to obtain when $N \lt 250$, so long as the two sides span an integer. So, we just have to march backward from $N=250$ and find the first place where the two sides do not span an integer.

We can see that the first (and greatest) failure occurs at $\boxed{N = 239}.$