Will there ever be a 5 homer game?

As many people probably know, the 4 home run game is one of the rarest feats in baseball.  There have been 16 4-homer games in baseball history, the last of which was Josh Hamilton's monster performance in 2012.  Here are the last 14 instances (since 1914) using the Baseball-Reference Play Index.  In all but one of these games, the player at least had a chance to hit a 5th homer.  Only Carlos Delgado's  4 HR game in 2003 came in exactly 4 plate appearances.  Will we ever see a player hit 5 homers in a game?  I crunched the numbers to find out our chances.

In order to spare people from all of the gory math details, I will try to simplify the explanation.  Let's start with a single player in a single game.  The important numbers we need to know are a player's average home run rate (HR per Plate Appearance) and how many plate appearances he received in the game.  For simplicity, we will assume that each plate appearance is an independent event, i.e. that the likelihood of a home run in one trip to the plate does not affect the next trip to the plate.  With these two numbers, we can determine the probability that he hits any given number of home runs in those plate appearances using the binomial distribution.  For example, let's assume that a player hits a homer in 3% of his plate appearances (the average for the league is currently about 2.8%).  If he gets exactly 4 plate appearances in a single game, his chances are approximately 1 in 1.23 million to homer in all 4 trips to the plate.  However, even giving him just one more chance (4 homers in 5 plate appearances) drastically improves his odds.  This somewhat average power hitter now has a 1 in 253 thousand chance to hit 4 homers in that single game.  If we give him 6 chances, the probability again improves to 1 in 86,400.  

If you thought those odds were bad, now consider the same player hitting 5 homers in 5 plate appearances.  The chances of that happening are 1 in 41.2 million.  But what about a hitter with more power?  A player with a 6% home run rate (for example, a guy like Mike Trout this season) has a 1 in 1.29 million chance to hit 5 homers in 5 chances, which is nearly the same chance that our "average" player has to hit 4 in 4 chances.  The two main conclusions we can make are:

1. A higher home run rate exponentially increases a player's chances of hitting 5 homers in a a game.  A great power hitter (6% home run rate) is 32 times as likely than an average power hitter (3% home run rate) to hit 5 HR in a game.
2. An extra plate appearance exponentially increases a player's chances of hitting 5 homers in a a game.  Hitting 5 homers in a game with 6 PAs is 5.85 times as likely as hitting 5 homers in a game with only 5 PAs.

In order to find our chances of any player hitting 5 home runs, we need to consider the league as a whole.  We cannot simply use the average home run rate and assume that every game played in a season features 9 "average" players on each team.  This would actually underestimate our true chances of seeing a 5 HR game.  Likewise, we cannot assume that the average player gets exactly 5 plate appearances in a game.  In both cases, we must consider a distribution of different numbers.  I used the top 250 hitters in terms of total plate appearances to find the distribution of home run rates throughout baseball, ranging from Giancarlo Stanton and Nelson Cruz to Michael Bourn and Eric Sogard.  In addition, I found the distribution of number of plate appearances per game.  It turns out that this eliminates almost 75% of all plate appearances since players are most likely to only have 4 plate appearances in a game (58% of all PAs come in 4 PA games, 16.6% of all PAs come in games with 3 PA or less).

Using all of this data, I created a formula to compute several different estimates given the current offensive environment in baseball:

• On average, we should see 9.63 3-homer games per year
• A 4 HR game should happen on average every 5.1 years (19.7% probability)
• A 5 HR game should happen on average every 405 years (0.247% probability)

So you're saying there's a chance?  Yes, every 405 years seems like a longshot but not everything happens at regular intervals.  We could get lucky.  Using these numbers, there is about a 10% chance that it happens in the next 40 years.  I just hope I am alive when it finally does.