Aaron Hill Rides the Bi-cycle

Please excuse the lame wordplay title (wordplay is fun!) and listen to this piece of truth: Aaron Hill made history last night. If your first questions is, "Tim, who the hell is Aaron Hill?", you can join most MLB fans in being duly surprised that a random role player is getting so much press. What is more surprising is that Mr. Hill has now hit for the cycle in two games in the same calender month. How rare is this? The only other player to do so hit two cycles in the same week in 1883. He is also only the second player since 1900 to hit two cycles in the same season (Babe Herman in 1931 for those of you keeping score at home). How rare is the cycle? If he hits one more IN HIS ENTIRE CAREER he will be tied for the most all time. Ludicrous.



Seeing as I am full of geekdom, a love for Excel, and WAY too much free time, lets look at the probability of this event occurring this season. A few assumptions will be made to simplify the problem but feel free to critique my method and give suggestions for how to change my model in the comment section below.

Firstly, I am not factoring in lineup position. This is simply not possible- as I see it- using my method. I do not have access to databases, that I know of, that would allow me to query and gather the needed information.  Secondly, I will be looking at this from the perspective of 3-6 plate appearances, but will be doing so using 2010 plate appearance data due to not having access to more recent data I know that this is not as rigorous as an analysis could be, but as before, the lack of better databases will limit my ability to analyze the information in an efficient manner. Also, considering the average PA/Game so far in 2012 is 4.13, I feel more comfortable in assuming this data spread. (And in case you are really wondering, Hill completed his feat in 4 PAs his first cycle and in his first 4 PAs of his second cycle game). Finally, I will only be using the 160 qualified hitters so far in 2012 for the batted ball data. This will give us a more liberal estimate, but it removes pitchers and other players that would not play enough to have a realistic shot at 4 PAs in a consistent manner.

With those assumptions out of the way, there are 24 permutations of getting a cycle. The most obvious in the natural cycle (which is WAY rare) at 1B-2B-3B-HR, but in a regular cycle, the order does not matter. The 2010 plate appearance data told me that hitters get 3 PAs 10.1% of the time, 4 PAs 59.1%, 5 PAs 27.4%, and 6+ 3.4%. Using a natural cycle calculation as a baseline (1B/PA * 2B/PA * 3B/PA * HR/PA), I found a natural cycle should occur in the following manner in 2012:

PA	% PA	Natural %	Weighted Probability
3	10.10%	0%	0.00%
4	59.10%	0.000127%	0.0000753%
5	27.40%	0.000637%	0.0001745%
6+	3.40%	0.001911%	0.0000650%
			0.0003148%

Ok, so clearly a natural cycle is not likely to happen this season. But Aaron Hill did not hit a natural cycle in either of his games. What about the probability of just one regular cycle happening? As mentioned above, there are 24 permutations for the cycle, so that gives the table below.

PA	% PA	Cycle %	Weighted Probability
3	10.10%	0%	0.00%
4	59.10%	0.003058%	0.0018071%
5	27.40%	0.015289%	0.0041891%
6+	3.40%	0.045866%	0.0015594%
			0.0075556%

Obviously, there is a greater chance for a regular cycle than a natural cycle. Even still, the chance of a cycle occurring is about 1 in 13,235. This works out to about 2.7 cycles per season. Considering we have seen more no-hitters than cycles the past few seasons, its a rare feat!

But what about Mr. Hill? Given his season output, is he more likely to get a cycle than the 160 qualified hitters?

His 2012 line looks something like this:

	1B	2B	3B	HR
% Chance/PA	16.44%	5.37%	1.34%	3.36%

This compares favorably to the 160 qualified hitters in 2012 who have a line of 15.68%, 4.88%, 0.54% and 3.09%. So if Hill has better even results for every category, what would his cycle chances be? Dear reader, you ask the questions (through me of course) and I will gladly answer with yet another table!

PA	% PA	Cycle %	Weighted Probability
3	10.10%	0%	0.00%
4	59.10%	0.009544%	0.0056404%
5	27.40%	0.047719%	0.0130750%
6+	3.40%	0.143157%	0.0048673%
			0.0235828%

This results in Aaron Hill hitting a cycle in every 4,240th game, or 1 every 26 seasons. Notice this is more often than the average player, where we expect it to happen about every 13,235th game, but I am comfortable saying this is one of those 'crazy awesome, statistic defying baseball events' that we see from time to time.

The fact that one player potentially hit the only two cycles this season is amazing.  Sure, a few more players might get the elusive cycle game,  but Aaron Hill's accomplishment is quite unique. If history is any precedent, we will probably never see this again in our lifetime.