Wednesday, July 3, 2013

The Ultimate No-Hitter

I know I haven't written much since baseball season got going, but something happened last night that gave me some inspiration.  As you might know, last night Reds pitcher Homer Bailey threw his second no-hitter in as many years.  I just barely caught the last out of the game on a live look-in on my phone because none of the eighteen ESPN channels was showing it.  In my haste to try to catch a bit of history, I drew the attention of my roommate, who has absolutely no interest in or knowledge of baseball.  He asked what was the big deal, and I said there was a no-hitter going on.  To which he responded something to the effect of, "Does that mean nobody hit a pitch?"

First of all, **sigh**.  I was raised in too baseball-centric a family to accept that kind of question.  Of course, I probably should have been less surprised since he famously asked when halftime was during a Phillies playoff game in '08 (man, do those days seem far off, huh).

But his question got me thinking -- what would it take for there to be an ACTUAL no-hitter?  No hits, no groundouts, no foul balls, no contact at all.  I knew it was essentially impossible, but I kind of wanted to see how impossible it really would be.  And, being the egotist I am, I'm going to share the experience with you.

My approach was to basically assume that everything about the game would be average, except that the batters would magically never hit a pitch they swung at.  This means that every plate appearance would either result in a strikeout or walk (let's leave the hit-by-pitch aside for this exercise, and it's rare enough that I feel OK doing it), additionally meaning that every plate appearance would last between 3 and 6 pitches.  To keep it classy, I'll just assume that that means that the average plate appearance takes up 4.5 pitches.

Given those assumptions, how many batters would the pitcher need to face if he had to complete a game with only walks and strikeouts?  Let's take the average rate of balls to strikes, and use a simple binomial probability distribution to estimate the probability of getting a walk (and, by extension, a strikeout) in this situation.  This season, there have been about 230,000 strikes and 130,000 balls thrown, but that's not the real ratio we would see in this situation.  We have to disregard foul balls, because players won't make contact in this game, and fouls increase the pitch count per at-bat beyond the maximum 6 if you don't make contact.  Other contact plays would just end up being swinging strikes that wouldn't mess up our expected rates.  I used the average rates of total swings, swings-and-misses, and balls in play to infer the number of foul balls to be about 66000, meaning that we would expect there to be about a 55/45 rate of strikes to balls.  Using a binomial distribution, I estimated the probability of getting 4 balls or more in an at-bat at 9.2%.

With that, we would expect about two walks over the 27 batters faced in a normal game, thus extending the game to 29 batters faced (yes, with 27 strikeouts -- your fantasy team would straight dominate).  With an average of 4.5 pitches per at-bat, that's an expected 131 pitches, which is actually not the most unreasonable amount (consider that everyone's boy Francisco Liriano threw a no-hitter with 8 walks and 149 pitches).

Of those 131 pitches, we would expect 55%, or 72 pitches, to be strikes.  The league average swing rate is 46%, and since that 46% must all be within that 55%, that means that the remaining strikes that weren't swung at would represent 9% of the total, or 12 pitches.  That leaves 60 pitches swung at and missed.  If the league average rate of contact on swings is 80%, that means that the odds of swinging and missing is 20%.  So, we could estimate the odds of swinging and missing at 60 pitches in a row at 0.2^60, or a daunting 1-in-1,000,000,000,000,000,000,000,000,000,000,000,000,000,000.  In other words, don't expect the ultimate no-hitter any time soon.