The odds of a cycle

The last 15 months have seen a resurgence in players hitting for the cycle. On Sunday, Pirates prospect Alex Presley became the 11th minor leaguer to hit for the cycle this year. Last season, eight players accomplished the feat—more than any other season except 1933.

There’s no doubt it’s rare, and anything so rare is going to have a hefty dash of flukiness. For every Mark Teixeira or Troy Tulowitzki, there’s a Fred Lewis or Brad Wilkerson (who did it twice!). Still, some hitters have a better chance than others.

Let’s walk through the steps of calculating the probability that any given player hits for the cycle.

Natural cycle

First, let’s simplify the problem. A “natural cycle” is a cycle in which the four hits come in progressive order: first the single, then the double, and so on. If a hitter is only given four plate appearances, there’s only one set of acceptable outcomes. He must hit the single in his first at-bat, the double in his second at-bat, etc.

Using Curtis Granderson‘s in-season CHONE projections, we can do the math for him. CHONE predicts he’ll hit a single in about 13.8 percent of plate appearances, a double in 4.5 percent, a triple in 1.6 percent and a home run in 3.8 percent. Multiply those together, and you have the probability that Granderson will hit all four, in order, in a given four-PA game. The result is 0.0000038, or about once in every 263,000 games.

Of course, Granderson wouldn’t be limited to four-PA games. Given five plate appearances, he is five times as likely to hit for a natural cycle. The plate appearance that isn’t part of the natural cycle could be the first, second, third, fourth or fifth of the game. So the probability jumps all the way up to 0.000019, or about once every 52,600 games.

It’s even easier in a six-PA game. In that case, there are 15 permutations of the four events of the natural cycle, so the probability is 0.000057, or about once every 17,500 games.

Given these numbers, it shouldn’t come as a surprise that only 14 players have ever hit for a natural cycle in a major league game!

We’re not quite done yet. To put together the odds of a four-PA natural cycle, a five-PA natural cycle and a six-PA natural cycle, we need to know how likely each number of plate appearances is.

(For today, we’re going to lump all games with six or more plate appearances into the “six” category. We’re also going to ignore the fact that when someone is in the process of hitting for the cycle, his team is slightly more likely to be scoring runs, so the probability of more plate appearances is greater.)

Thanks to some quick-and-dirty querying courtesy of THT Captain Dave Studeman, we know that in 2010, the average starter gets three plate appearances about 10.1 percent of the time, four PAs in 59.1 percent of games, five PAs 27.4 percent of the time, and six (or more) PAs in 3.4 percent of games. Of course that varies greatly depending on lineup spot, and we’ll revisit the importance of that later. It also assumes that a player stays in the game—a reasonable assumption if he’s getting close to a cycle!

Thus, if we know nothing about where Granderson will hit in the lineup, we can calculate his odds of a natural cycle as follows:

PA     PA%      NC%    Result  
3   10.10%       0%        0%  
4   59.10%  0.0004%  0.00022%  
5   27.40%  0.0019%  0.00052%  
6    3.40%  0.0057%  0.00019%  
Total                0.00094%

In other words, if you plopped a Granderson clone into a simulator full of league-average pitchers and parks, he’d come up with a natural cycle about once every 100,000 games, or about once every 650 seasons.

Unnatural cycles

The natural cycle is so rare that it’s not of much interest in itself. Thanks to our work so far, we’ve done most of the heavy lifting, and we can apply the results to calculating the odds of “normal” cycles, ones in which the four hits don’t have to come in any particular order.

There are 24 permutations of the sequence “single, double, triple, home run.” Thus, a garden-variety cycle is 24 times more likely than a natural cycle. Using the numbers we came up with above, Granderson would hit for the cycle about once every 4,400 games, or once per 27 seasons.

To get a general idea of the likelihood of cycles, let’s replace Granderson’s numbers with league-average figures. Last year, the average hitter singled 15.4 percent of the time, doubled 4.7 percent of the time, tripled 0.5 percent of the time, and homered 2.7 percent of the time.

Here’s the same chart as above, only using the odds of any cycle along with 2009 league average hitting stats:

PA        PA%       C%       Result  
3      10.10%       0%           0%  
4      59.10%  0.0024%     0.00142%  
5      27.40%  0.0120%     0.00329%  
6       3.40%  0.0350%     0.00119%  
Total                      0.00590%

The resulting percentage is equivalent to about one cycle per 17,000 player-games. When 30 teams play a 162-game season, that’s 43,740 player-games, meaning we should expect about 2.5 cycles per season. (At least in 2009′s run environment. More extra-base hits means more cycles.) In recent memory, the average has been higher than that—the last time there were fewer than three cycles in a season was 2002.

The importance of lineup position

For just about any single-game offensive achievement, lineup position matters. The higher you bat in the lineup, the more times you’ll hit, and the more times you hit, the more likely you are to do something good.

As we discovered earlier, the odds of hitting for the cycle are five times greater with five plate appearances instead of four; they are 15 times greater with six plate appearances instead of four. And obviously, if you only get three plate appearances, you aren’t going to hit for the cycle. That isn’t a factor at the top of the order, but if you’re batting ninth, it’s a very real possibility.

Once again, Curtis Granderson makes for a great example. A year ago, we knew him as a leadoff hitter; now, he’s usually near the bottom of the lineup. Here’s a look at how likely he is to hit for the cycle given his position in the lineup. “Games” refers to how often he is likely to hit for the cycle (e.g., once per 2,651 games), and “Seasons” is another way of describing the same thing (e.g., once per 16 seasons).

Spot  p(Cycle)  Games  Seasons  
1       0.038%   2651       16  
2       0.033%   3037       19  
3       0.029%   3491       22  
4       0.025%   4002       25  
5       0.021%   4719       29  
6       0.018%   5522       34  
7       0.016%   6415       40  
8       0.013%   7651       47  
9       0.011%   9275       57

The difference is striking. Moving from the leadoff spot to the seventh position more than halves Granderson’s chances of hitting for the cycle in any given game.

The 2009 cyclists

As I’ve noted, last season was a banner year for cycles. Every cycle is a surprise, but some are more surprising than others.

Here are the eight 2009 cyclists. I show two probabilities for each one. First, lineup-agnostic odds using their 2009 CHONE projections. This is the probability that the player hits for a cycle at some point during a season in which he starts 150 games. Next, I show the odds of achieving the feat from his specific lineup spot that day.

Player           Cycle150  Lineup  LU-150  
Jason Kubel         1.56%       4   1.72%  
Felix Pie           1.54%       7   1.07%  
Troy Tulowitzki     1.23%       4   1.36%  
Orlando Hudson      0.99%       2   1.44%  
Michael Cuddyer     0.97%       4   1.07%  
Ian Kinsler         0.96%       1   1.61%  
Melky Cabrera       0.91%       6   0.73%  
B.J. Upton          0.86%       6   0.69%

Jason Kubel came into the season with the best chance of hitting for a cycle, and batting in the cleanup spot didn’t make it any easier. By contrast, Ian Kinsler ranked behind nearly 200 other hitters in cycle-chances, yet leading off for the Rangers boosted his odds quite a bit.

Who’s next?

So far, the 2010 season has been a letdown compared with last year. Jody Gerut is the only player who has managed a cycle.

Returning to the in-season CHONE projections, we can figure the odds of any given player hitting for the cycle. Here are the top 30. Make sure to scan for the bottom: The best part of ranking obscure stats is the appearance of an absolute surprise.

Player             Team  Cycle150  
Gonzalez Carlos     COL      3.6%  
Wise Dewayne        PHI      3.5%  
Granderson Curtis   NYA      3.3%  
Kemp Matt           LAN      3.0%  
Jones Adam          BAL      3.0%  
Braun Ryan          MIL      3.0%  
Bruce Jay           CIN      2.9%  
Upton Justin        ARI      2.9%  
Crawford Carl       TBA      2.5%  
Torres Andres       SFN      2.4%  
Schierholtz Nate    SFN      2.4%  
Sandoval Pablo      SFN      2.3%  
Olivo Miguel        COL      2.3%  
Tulowitzki Troy     COL      2.3%  
Pagan Angel         NYN      2.3%  
Pence Hunter        HOU      2.2%  
Rollins Jimmy       PHI      2.2%  
Hart Corey          MIL      2.2%  
Victorino Shane     PHI      2.2%  
Pie Felix           BAL      2.1%  
Brown Domonic       PHI      2.1%  
Drew Stephen        ARI      2.1%  
Raburn Ryan         DET      2.0%  
Colvin Tyler        CHN      2.0%  
Reyes Jose          NYN      2.0%  
Davis Leonard       WAS      2.0%  
Heyward Jason       ATL      2.0%  
Maybin Cameron      FLO      2.0%  
Taylor Michael      OAK      1.9%  
McCutchen Andrew    PIT      1.9%

Once again, the percentages are each player’s chances of hitting for the cycle in 150 starts, without taking lineup position into account. If a player is leading off, you can multiply his chances by about 1.7; if he’s hitting ninth, his chances are half what is listed here.

The inclusion of a triple in a cycle explains why so many of the players who have accomplished the feat aren’t big stars. The same holds true for many on this list. CHONE projects Dewayne Wise for three triples in under 200 plate appearances—unlike for most players, the triple requirement isn’t going to hold him back. The same flukiness puts Nationals minor leaguer Leonard Davis on the list despite his .246/.292/.415 slash line.

Despite the surprises, it helps to be good. After all, the list of players who have hit for multiple cycles includes such luminaries as George Brett, Pete Browning, Mickey Cochrane, Joe DiMaggio, Bobby Doerr, Lou Gehrig, George Sisler and Arky Vaughan. Add Matt Kemp, Ryan Braun or Jimmy Rollins to the list and he wouldn’t be out of place.

Add Andres Torres or Ryan Raburn to the list and … well, even Chad Moeller hit for the cycle once.

References & Resources
The wikipedia page on the cycle has more trivia than you could possibly desire. Also helpful is Retrosheet’s chronological list of cycles, linking to boxscores.

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Comments

  1. Dave Studeman said...

    Good stuff, Jeff.  At the top of the article, you say that Presley is the 11th minor leaguer to hit for the cycle this year.  Did you mean that?

  2. studes said...

    Okay, then I think there’s a mistake in your opening paragraph.  You say that there has been a cycle 11 times this year, then you say that it was done eight times last year, the most since 1933.  I assume that you meant to say “major leagues” in the second instance?

  3. Brian said...

    I think the odds are higher than what you’ve figured out.  On certain days there will either be a terrible pitcher throwing, favorable weather conditions, etc., conditions which would greatly increase the probability.  Of course the opposite is true, but with the starting probability near zero already, this is a lesser effect.  This is why we’ve seen more cycles in recent seasons.

  4. Randy said...

    Could see Torres or Sandoval going the cycle but kind of surprised to see three GIANTS hitters on the list…

    How important is the home park when it comes to CHONE projections? (I guess for cycles it may be more important that it is relatively easy to hit triples at AT&T than that it is relatively difficult to hit homers)

  5. Jeff Sackmann said...

    Brian – you’re right.  There are many, many reasons why the probability on any given day would be higher than using season lines against average pitching.  I also suspect that if you’re close to hitting for a cycle, you’re more likely to get 5 or 6 PAs than usual.

    Randy – CHONE is park-adjusted, so those probabilities take into account AT&T.  I don’t know about specific triples factors, but it’s certainly true that the triple is a bigger deal than the homer—that’s why DeWayne Wise is ranked second!

  6. David said...

    A weird thing last week was when the Mets playing Twins on 6/30 hit for the cycle in the course of 5 AB in the 5th inning. There’s probably some team that hit for the cycle in 4 AB.

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