Season's Cycle

Courtesy of Mike's Baseball Rants.

Brad Wilkerson's sequential cycle got me to thinking about the odds of sequential cycles (single, double, triple, and home run in order) and cycles in general. After going over the probabilities, Wilkerson's feat seems even more remarkable to me.

In the world of probabilities, a four-at-bat, sequential cycle is the rarest feat. Why? It's all probability—probability is our friend. Hitting a sequential cycle in 4 at-bats is like rolling a die and getting 1, 2, 3, and 4 in order, except that the die would have to be a bit odd-sided since the probability of hitting a single or of not getting a hit at all are much higher than getting a triple or home run.

Historically, the odds of getting a hit are not great. Here are the odds for each type of hit based on total plate appearances across major-league history (TPA= AB + BB + HBP + SF + SH):

Single: 17.10% Double: 3.83% Triple: 0.88% Home Run: 1.60%

Today doubles and home runs are more plentiful and singles and triples are rarer than the historical average (Actually a league has not recorded a single or triple percentage as high as the average since the mid-Forties).

So unlike a die, on which the odds of getting a one are the same as the odds of getting any other number, the odds are much, much higher of not getting a hit and, then if a hit is made, of collecting a single. To calculate the odds of getting a sequential cycle in four at-bats would be the same as with dice except instead of using 1-to-6 as the odds, you would insert the odds above (for a historical average). So whereas a 1-2-3-4 roll would be 1/6 * 1/6 * 1/6 * 1/6 or about 0.00077 (7.7 * 10^4) , the odds of a sequential cycle would be 17.10%* 3.83%*0.88%*1.60% or about 0.00000092 (9.2 * 10^7). Given that there have been 179277 games in major league history each with two teams of nine batters, the expected number of 4-at-bat, sequential cycles in major-league history is just 2.97.

The previous sequential cycler was Jose Valentin in 2000, but he did it in a 5-at-bat game. The odds of a 5-at-bat, sequential cycle go up somewhat. The at-bat that was not part of the sequential cycle would have to be before or after the other four hits, and it wouldn't matter what he did in that plate appearance. Using the die example, there are two scenarios X-1-2-3-4 or 1-2-3-4-X (where X is the mystery at-bat). Therefore the odds increase two-fold.

There have been thirteen sequential cycles according to Baseball-Almanac.com including Wilkerson's:

Name	Team	Date
Bill Collins	Boston (NL)	10-06-1910
Bob Fothergill	Detroit	09-26-1926
Tony Lazzeri	New York (AL)	06-03-1932
Charlie Gehringer	Detroit	05-27-1939
Leon Culberson	Boston (AL)	07-03-1943
Jim Hickman	New York (NL)	08-07-1963
Ken Boyer	St. Louis (NL)	06-16-1964
Billy Williams	Chicago (NL)	07-17-1966
Tim Foli	Montreal	04-22-1976
Bob Watson	Boston (AL)	09-15-1979
John Mabry	St. Louis (NL)	05-18-1996
Jose Valentin	Chicago (AL)	04-27-2000
Brad Wilkerson	Montreal	06-24-2003

I don't think even Retrosheet can help us determine if that 2.97 expected value is in the ballpark or not. We are left with old Sporting News microfilm. I think I'll pass for now.

Next in the cycle food chain, you have the four at-bat, not-necessarily-sequential variety. Let's revisit the die: Let's assume that we throw a one first (to limit the results). To complete a cycle in 4 tries, here are the possibilities:

1-2-3-4 1-2-4-3 1-3-2-4 1-3-4-2 1-4-2-3 1-4-3-2

If my fancy ciphering works, that's six. Now, consider that given that there are four ways to start that run (1, 2, 3, or 4), there are 24 possibilities (4 * 6 or 4!—four factorial—if you prefer). That means the odds of a four-at-bat cycle historically are 0.0022% which translates into 59.68 expected cycles all-time, if all games were based on 4 at-bats. But they're not

In a 5-at-bat (or 5-plate-appearance to be exact) game, the odds go up five-fold. Again using the die, if we had five throws to collect 1, 2, 3, and 4 in any order, that would mean that we would have one "free" throw that could be anything: a hit, an out, an interference call, a Jeffrey Maier catch—who cares? Using just the first example above we could have five combinations to achieve the cycle:

X-1-2-3-4 1- X-2-3-4 1-2- X-3-4 1-2-3- X-4 1-2-3-4-X (Where X is the "free" throw

So each 4-at-bat combination now propagates to five 5-AB combinations or that hitting for the cycle in 5 plate appearances is five times as likely then in four. This means that there 120 combinations that can result in a cycle (i.e., five times the 24 from 4-ABs or 5!)

A six-at-bat game is 25 times more likely to result in a cycle than a 4-AB game since there are two "free" throws. Here are the resulting 6-AB combinations from just the first 5-AB combination above:

X- X-1-2-3-4 X-1- X-2-3-4 X-1-2- X-3-4 X-1-2-3- X-4 X-1-2-3-4-X (Where X is the "free" throw

Therefore, there are five 6-AB combinations for every 5-AB one and 25 for every 4-AB one. That means that there are 600 combinations in a 6-AB game that can result in a cycle (5 * 5 * 4!).

Also, I should mention that the odds of each combination do not change because the odds of getting anything in the "free" throw are 1. If you throw the die, you have to get something. Such a certainty is assigned the highest probability, one. So one times the combination percentage is still the original percentage.

So what does this all mean, if anything? It means that we can take the probability of hitting a single, double, triple, and home run for each league year and using the number of games, determine the expected occurrences of a cycle. Then we can see how they compare with the actual totals.

At the risk of overkill, I now list the odds of hitting for the cycle and the expected total of cycles per league and year based on 4-, 5-, and 6-plate-apperance at-bats. Also, I include an "Avg Exp" column which calculates the expected number of cycles based on the actual average of plate appearances per game (usually around 4.25). Lastly, I list the actual times a batter hit for the cycle for each league-year. (Note that I could only find NL and AL data. Data based on Baseball-Almanac.com's cycle data):

year	Lg	Cycle Prob (4PA)	Exp Per Season	Cycle Prob (5PA)	Exp Per Season	Cycle Prob (6PA)	Exp Per Season	Avg Exp	Actual
1871	NA	0.0018%	0.0403	0.0088%	0.2013	0.0440%	1.0064	0.222608
1872	NA	0.0004%	0.0127	0.0019%	0.0635	0.0096%	0.3175	0.065704
1873	NA	0.0006%	0.0210	0.0029%	0.1051	0.0147%	0.5255	0.108099
1874	NA	0.0004%	0.0153	0.0018%	0.0767	0.0092%	0.3835	0.063751
1875	NA	0.0002%	0.0144	0.0012%	0.0718	0.0058%	0.3592	0.040303
1876	NL	0.0003%	0.0132	0.0014%	0.0659	0.0070%	0.3295	0.037637
1877	NL	0.0004%	0.0130	0.0020%	0.0648	0.0100%	0.3241	0.034016
1878	NL	0.0003%	0.0088	0.0013%	0.0438	0.0066%	0.2188	0.018793
1879	NL	0.0006%	0.0319	0.0028%	0.1597	0.0138%	0.7985	0.074807
1880	NL	0.0006%	0.0341	0.0028%	0.1703	0.0139%	0.8517	0.049681
1881	NL	0.0007%	0.0420	0.0035%	0.2101	0.0174%	1.0505	0.084333
1882	AA	0.0005%	0.0223	0.0027%	0.1117	0.0133%	0.5583	0.039646
1882	NL	0.0014%	0.0840	0.0069%	0.4200	0.0345%	2.0999	0.180159	1
1883	AA	0.0009%	0.0628	0.0045%	0.3138	0.0224%	1.5691	0.142557
1883	NL	0.0014%	0.1007	0.0071%	0.5036	0.0354%	2.5182	0.220652
1884	AA	0.0011%	0.1363	0.0057%	0.6815	0.0287%	3.4075	0.205126
1884	NL	0.0017%	0.1412	0.0086%	0.7060	0.0429%	3.5301	0.278871	1
1884	UA	0.0006%	0.0465	0.0030%	0.2324	0.0151%	1.1619	0.06931
1885	AA	0.0010%	0.0771	0.0048%	0.3856	0.0241%	1.9282	0.153052
1885	NL	0.0010%	0.0785	0.0049%	0.3923	0.0245%	1.9615	0.123907	2
1886	AA	0.0008%	0.0773	0.0039%	0.3864	0.0193%	1.9321	0.166585
1886	NL	0.0015%	0.1327	0.0074%	0.6633	0.0372%	3.3167	0.186939	2
1887	AA	0.0015%	0.1450	0.0073%	0.7249	0.0366%	3.6247	0.369867
1887	NL	0.0029%	0.2690	0.0147%	1.3449	0.0735%	6.7244	0.6052	1
1888	AA	0.0008%	0.0750	0.0038%	0.3752	0.0190%	1.8762	0.138463
1888	NL	0.0013%	0.1306	0.0067%	0.6529	0.0333%	3.2646	0.17719	2
1889	AA	0.0015%	0.1486	0.0074%	0.7431	0.0369%	3.7157	0.359337
1889	NL	0.0016%	0.1543	0.0081%	0.7715	0.0404%	3.8577	0.414233	2
1890	AA	0.0008%	0.0779	0.0040%	0.3895	0.0200%	1.9473	0.172937
1890	NL	0.0012%	0.1158	0.0060%	0.5790	0.0298%	2.8951	0.293713	2
1890	PL	0.0020%	0.1938	0.0102%	0.9690	0.0509%	4.8449	0.695285
1891	AA	0.0012%	0.1236	0.0061%	0.6181	0.0307%	3.0907	0.331337
1891	NL	0.0013%	0.1287	0.0065%	0.6433	0.0324%	3.2166	0.352685	1
1892	NL	0.0010%	0.1650	0.0050%	0.8248	0.0249%	4.1240	0.379045
1893	NL	0.0019%	0.2664	0.0094%	1.3321	0.0471%	6.6603	0.941402
1894	NL	0.0042%	0.6030	0.0211%	3.0148	0.1056%	15.0742	1.966558	4
1895	NL	0.0022%	0.3186	0.0111%	1.5932	0.0556%	7.9659	0.98869	2
1896	NL	0.0018%	0.2521	0.0088%	1.2606	0.0442%	6.3030	0.711935	2
1897	NL	0.0016%	0.2320	0.0080%	1.1598	0.0398%	5.7989	0.620775
1898	NL	0.0008%	0.1275	0.0038%	0.6376	0.0192%	3.1881	0.267988
1899	NL	0.0011%	0.1827	0.0055%	0.9133	0.0275%	4.5667	0.378746
1900	NL	0.0013%	0.1310	0.0064%	0.6551	0.0320%	3.2753	0.291997
1901	AL	0.0015%	0.1464	0.0074%	0.7318	0.0370%	3.6588	0.341918	2
1901	NL	0.0010%	0.1026	0.0051%	0.5130	0.0254%	2.5651	0.230173	1
1902	AL	0.0015%	0.1485	0.0075%	0.7424	0.0373%	3.7119	0.314118
1902	NL	0.0003%	0.0300	0.0015%	0.1502	0.0074%	0.7511	0.054234
1903	AL	0.0011%	0.1068	0.0054%	0.5342	0.0268%	2.6711	0.169472	3
1903	NL	0.0007%	0.0726	0.0036%	0.3632	0.0180%	1.8158	0.151138	1
1904	AL	0.0006%	0.0687	0.0031%	0.3437	0.0153%	1.7185	0.086951
1904	NL	0.0006%	0.0672	0.0030%	0.3361	0.0150%	1.6803	0.083364	2
1905	AL	0.0005%	0.0550	0.0025%	0.2749	0.0124%	1.3745	0.077556
1905	NL	0.0007%	0.0774	0.0035%	0.3868	0.0173%	1.9340	0.125049
1906	AL	0.0005%	0.0504	0.0023%	0.2519	0.0114%	1.2596	0.069312
1906	NL	0.0004%	0.0404	0.0018%	0.2018	0.0091%	1.0092	0.047863
1907	AL	0.0003%	0.0324	0.0015%	0.1618	0.0073%	0.8089	0.04675
1907	NL	0.0004%	0.0436	0.0020%	0.2178	0.0098%	1.0892	0.042664	1
1908	AL	0.0003%	0.0367	0.0016%	0.1833	0.0082%	0.9166	0.046227	1
1908	NL	0.0004%	0.0467	0.0021%	0.2337	0.0104%	1.1683	0.054851
1909	AL	0.0003%	0.0358	0.0016%	0.1789	0.0080%	0.8944	0.040408
1909	NL	0.0004%	0.0490	0.0022%	0.2452	0.0110%	1.2261	0.07879
1910	AL	0.0005%	0.0511	0.0023%	0.2554	0.0113%	1.2769	0.074165	1
1910	NL	0.0008%	0.0886	0.0040%	0.4429	0.0198%	2.2145	0.160811	2
1911	AL	0.0009%	0.1013	0.0046%	0.5065	0.0229%	2.5323	0.213017	1
1911	NL	0.0014%	0.1544	0.0069%	0.7722	0.0344%	3.8608	0.314521	1
1912	AL	0.0007%	0.0803	0.0036%	0.4014	0.0180%	2.0070	0.148836	2
1912	NL	0.0015%	0.1623	0.0074%	0.8116	0.0368%	4.0580	0.353344	2
1913	AL	0.0007%	0.0728	0.0033%	0.3639	0.0165%	1.8195	0.109891
1913	NL	0.0014%	0.1523	0.0068%	0.7615	0.0341%	3.8074	0.271772
1914	AL	0.0005%	0.0588	0.0026%	0.2942	0.0130%	1.4710	0.084899
1914	FL	0.0013%	0.1430	0.0064%	0.7149	0.0318%	3.5744	0.229664
1914	NL	0.0009%	0.1031	0.0046%	0.5155	0.0229%	2.5777	0.157228
1915	AL	0.0006%	0.0625	0.0028%	0.3127	0.0140%	1.5633	0.111898
1915	FL	0.0010%	0.1083	0.0049%	0.5415	0.0243%	2.7073	0.173798
1915	NL	0.0008%	0.0948	0.0042%	0.4738	0.0211%	2.3691	0.13522	1
1916	AL	0.0005%	0.0547	0.0024%	0.2737	0.0122%	1.3687	0.09728
1916	NL	0.0009%	0.0989	0.0044%	0.4947	0.0221%	2.4735	0.140297
1917	AL	0.0004%	0.0475	0.0021%	0.2374	0.0106%	1.1868	0.08519
1917	NL	0.0008%	0.0848	0.0038%	0.4240	0.0188%	2.1199	0.124742
1918	AL	0.0004%	0.0330	0.0018%	0.1651	0.0090%	0.8255	0.063136
1918	NL	0.0006%	0.0548	0.0030%	0.2741	0.0150%	1.3706	0.089328	1
1919	AL	0.0011%	0.1156	0.0057%	0.5781	0.0287%	2.8907	0.242959
1919	NL	0.0009%	0.0877	0.0044%	0.4383	0.0218%	2.1917	0.13943
1920	AL	0.0019%	0.2105	0.0095%	1.0523	0.0474%	5.2617	0.521734	2
1920	NL	0.0012%	0.1318	0.0059%	0.6589	0.0297%	3.2945	0.278258	1
1921	AL	0.0028%	0.3113	0.0140%	1.5564	0.0702%	7.7819	0.907552	2
1921	NL	0.0026%	0.2870	0.0130%	1.4352	0.0650%	7.1759	0.653831	2
1922	AL	0.0025%	0.2795	0.0126%	1.3975	0.0628%	6.9876	0.720466	2
1922	NL	0.0028%	0.3166	0.0142%	1.5831	0.0709%	7.9156	0.814403	2
1923	AL	0.0020%	0.2179	0.0098%	1.0893	0.0491%	5.4467	0.580785
1923	NL	0.0025%	0.2807	0.0126%	1.4035	0.0632%	7.0177	0.743627	1
1924	AL	0.0019%	0.2160	0.0097%	1.0799	0.0486%	5.3997	0.59046	2
1924	NL	0.0026%	0.2855	0.0129%	1.4276	0.0646%	7.1381	0.647851
1925	AL	0.0026%	0.2861	0.0129%	1.4304	0.0645%	7.1519	0.834078	1
1925	NL	0.0035%	0.3882	0.0176%	1.9410	0.0881%	9.7049	1.015812	2
1926	AL	0.0021%	0.2280	0.0103%	1.1400	0.0514%	5.7000	0.596795	1
1926	NL	0.0021%	0.2379	0.0107%	1.1893	0.0535%	5.9465	0.526949
1927	AL	0.0024%	0.2642	0.0119%	1.3212	0.0593%	6.6061	0.679158
1927	NL	0.0021%	0.2326	0.0105%	1.1629	0.0524%	5.8147	0.555107	2
1928	AL	0.0026%	0.2901	0.0131%	1.4503	0.0653%	7.2516	0.71539	1
1928	NL	0.0025%	0.2815	0.0127%	1.4077	0.0637%	7.0384	0.752165	1
1929	AL	0.0031%	0.3433	0.0156%	1.7166	0.0778%	8.5830	0.914429	2
1929	NL	0.0038%	0.4221	0.0190%	2.1107	0.0952%	10.5537	1.214627	1
1930	AL	0.0040%	0.4412	0.0199%	2.2062	0.0995%	11.0310	1.233939
1930	NL	0.0052%	0.5810	0.0261%	2.9050	0.1306%	14.5252	1.731899	3
1931	AL	0.0027%	0.2980	0.0134%	1.4898	0.0670%	7.4490	0.860449
1931	NL	0.0024%	0.2626	0.0118%	1.3129	0.0590%	6.5646	0.608281	3
1932	AL	0.0033%	0.3673	0.0166%	1.8366	0.0830%	9.1829	1.103105	2
1932	NL	0.0030%	0.3296	0.0148%	1.6479	0.0741%	8.2395	0.82822
1933	AL	0.0026%	0.2880	0.0132%	1.4402	0.0658%	7.2010	0.839827	4
1933	NL	0.0016%	0.1745	0.0078%	0.8723	0.0392%	4.3615	0.343051	4
1934	AL	0.0026%	0.2825	0.0128%	1.4125	0.0638%	7.0627	0.819064	3
1934	NL	0.0026%	0.2817	0.0129%	1.4086	0.0644%	7.0429	0.706709
1935	AL	0.0029%	0.3167	0.0144%	1.5833	0.0720%	7.9167	0.967972
1935	NL	0.0026%	0.2869	0.0129%	1.4345	0.0646%	7.1724	0.708224	1
1936	AL	0.0036%	0.3985	0.0179%	1.9924	0.0896%	9.9621	1.285434
1936	NL	0.0021%	0.2395	0.0107%	1.1974	0.0536%	5.9871	0.646781	1
1937	AL	0.0036%	0.3997	0.0178%	1.9984	0.0892%	9.9920	1.135609	3
1937	NL	0.0023%	0.2524	0.0114%	1.2621	0.0568%	6.3107	0.572566
1938	AL	0.0033%	0.3686	0.0167%	1.8429	0.0835%	9.2144	1.058866	1
1938	NL	0.0022%	0.2366	0.0108%	1.1829	0.0539%	5.9143	0.576777
1939	AL	0.0031%	0.3409	0.0154%	1.7043	0.0770%	8.5216	0.983026	2
1939	NL	0.0022%	0.2432	0.0110%	1.2161	0.0548%	6.0807	0.583133	1
1940	AL	0.0035%	0.3922	0.0176%	1.9609	0.0880%	9.8043	1.04368	3
1940	NL	0.0021%	0.2377	0.0107%	1.1883	0.0535%	5.9413	0.580667	3
1941	AL	0.0027%	0.3021	0.0135%	1.5106	0.0675%	7.5528	0.825334	1
1941	NL	0.0015%	0.1706	0.0076%	0.8532	0.0381%	4.2661	0.402162
1942	AL	0.0015%	0.1652	0.0075%	0.8262	0.0376%	4.1309	0.400214
1942	NL	0.0011%	0.1263	0.0057%	0.6317	0.0286%	3.1584	0.279764
1943	AL	0.0011%	0.1254	0.0056%	0.6268	0.0282%	3.1341	0.3001	1
1943	NL	0.0011%	0.1283	0.0057%	0.6413	0.0287%	3.2063	0.291181
1944	AL	0.0013%	0.1469	0.0066%	0.7343	0.0330%	3.6715	0.353472	2
1944	NL	0.0016%	0.1818	0.0081%	0.9088	0.0405%	4.5440	0.431207	1
1945	AL	0.0012%	0.1270	0.0058%	0.6349	0.0288%	3.1744	0.282415
1945	NL	0.0014%	0.1526	0.0069%	0.7632	0.0343%	3.8161	0.39491	2
1946	AL	0.0018%	0.1994	0.0089%	0.9970	0.0446%	4.9851	0.448439	2
1946	NL	0.0014%	0.1590	0.0071%	0.7949	0.0356%	3.9745	0.362009
1947	AL	0.0018%	0.1990	0.0089%	0.9948	0.0444%	4.9740	0.442026	2
1947	NL	0.0024%	0.2698	0.0121%	1.3489	0.0604%	6.7444	0.649295
1948	AL	0.0021%	0.2372	0.0107%	1.1861	0.0533%	5.9306	0.643899	1
1948	NL	0.0022%	0.2498	0.0112%	1.2490	0.0560%	6.2450	0.576458	2
1949	AL	0.0019%	0.2084	0.0094%	1.0420	0.0468%	5.2098	0.554963
1949	NL	0.0023%	0.2626	0.0117%	1.3131	0.0586%	6.5653	0.639076	2
1950	AL	0.0026%	0.2949	0.0132%	1.4743	0.0661%	7.3715	0.8357	3
1950	NL	0.0027%	0.3051	0.0137%	1.5255	0.0686%	7.6273	0.748436	2
1951	AL	0.0019%	0.2165	0.0097%	1.0824	0.0487%	5.4118	0.573655
1951	NL	0.0024%	0.2661	0.0119%	1.3305	0.0594%	6.6524	0.637455	1
1952	AL	0.0017%	0.1845	0.0083%	0.9223	0.0413%	4.6117	0.448021	1
1952	NL	0.0019%	0.2168	0.0097%	1.0838	0.0487%	5.4190	0.451174
1953	AL	0.0020%	0.2195	0.0099%	1.0973	0.0493%	5.4867	0.540853
1953	NL	0.0033%	0.3649	0.0163%	1.8243	0.0815%	9.1217	0.833009
1954	AL	0.0019%	0.2121	0.0095%	1.0607	0.0474%	5.3036	0.503653
1954	NL	0.0029%	0.3258	0.0147%	1.6292	0.0735%	8.1459	0.825405	1
1955	AL	0.0019%	0.2060	0.0093%	1.0301	0.0463%	5.1504	0.51891
1955	NL	0.0028%	0.3099	0.0140%	1.5494	0.0699%	7.7470	0.700859
1956	AL	0.0022%	0.2462	0.0111%	1.2310	0.0553%	6.1551	0.659751
1956	NL	0.0028%	0.3088	0.0138%	1.5442	0.0691%	7.7211	0.62465
1957	AL	0.0019%	0.2106	0.0095%	1.0530	0.0475%	5.2652	0.509954	1
1957	NL	0.0027%	0.2989	0.0134%	1.4947	0.0671%	7.4734	0.743504	1
1958	AL	0.0019%	0.2105	0.0094%	1.0527	0.0472%	5.2634	0.426234
1958	NL	0.0028%	0.3133	0.0141%	1.5663	0.0706%	7.8317	0.728587
1959	AL	0.0018%	0.1984	0.0089%	0.9918	0.0446%	4.9590	0.450946
1959	NL	0.0025%	0.2762	0.0124%	1.3808	0.0619%	6.9040	0.616167	1
1960	AL	0.0018%	0.2028	0.0091%	1.0142	0.0457%	5.0711	0.489415	1
1960	NL	0.0025%	0.2818	0.0126%	1.4090	0.0632%	7.0451	0.615789	1
1961	AL	0.0022%	0.3166	0.0108%	1.5832	0.0542%	7.9160	0.741126
1961	NL	0.0027%	0.3035	0.0136%	1.5174	0.0681%	7.5868	0.670352	1
1962	AL	0.0022%	0.3166	0.0109%	1.5831	0.0544%	7.9155	0.766586	1
1962	NL	0.0023%	0.3354	0.0115%	1.6770	0.0574%	8.3852	0.758681
1963	AL	0.0018%	0.2612	0.0090%	1.3060	0.0449%	6.5302	0.558619
1963	NL	0.0018%	0.2649	0.0091%	1.3247	0.0454%	6.6235	0.487814	2
1964	AL	0.0017%	0.2530	0.0086%	1.2650	0.0432%	6.3250	0.529883	2
1964	NL	0.0020%	0.2866	0.0098%	1.4329	0.0490%	7.1647	0.551391	2
1965	AL	0.0017%	0.2427	0.0083%	1.2136	0.0416%	6.0678	0.479346	1
1965	NL	0.0020%	0.2862	0.0098%	1.4310	0.0489%	7.1548	0.584429
1966	AL	0.0019%	0.2709	0.0093%	1.3546	0.0467%	6.7731	0.493613
1966	NL	0.0021%	0.3050	0.0105%	1.5249	0.0524%	7.6244	0.620385	2
1967	AL	0.0014%	0.2036	0.0070%	1.0178	0.0349%	5.0892	0.36748
1967	NL	0.0017%	0.2517	0.0086%	1.2587	0.0432%	6.2934	0.497863
1968	AL	0.0012%	0.1701	0.0058%	0.8504	0.0291%	4.2520	0.264977	1
1968	NL	0.0011%	0.1666	0.0057%	0.8331	0.0285%	4.1654	0.291001
1969	AL	0.0014%	0.2405	0.0069%	1.2023	0.0343%	6.0116	0.529682
1969	NL	0.0016%	0.2888	0.0082%	1.4442	0.0412%	7.2209	0.6068
1970	AL	0.0015%	0.2652	0.0076%	1.3259	0.0379%	6.6297	0.580057	2
1970	NL	0.0023%	0.4105	0.0117%	2.0527	0.0587%	10.2636	1.019498	3
1971	AL	0.0013%	0.2180	0.0063%	1.0901	0.0313%	5.4503	0.429844	1
1971	NL	0.0016%	0.2743	0.0078%	1.3717	0.0392%	6.8583	0.570289
1972	AL	0.0010%	0.1621	0.0048%	0.8105	0.0242%	4.0523	0.271334	2
1972	NL	0.0015%	0.2570	0.0077%	1.2850	0.0384%	6.4250	0.552151	2
1973	AL	0.0016%	0.2840	0.0081%	1.4200	0.0406%	7.1002	0.644455
1973	NL	0.0015%	0.2622	0.0075%	1.3111	0.0375%	6.5554	0.595273	1
1974	AL	0.0014%	0.2519	0.0072%	1.2597	0.0360%	6.2984	0.528123
1974	NL	0.0014%	0.2524	0.0072%	1.2620	0.0361%	6.3100	0.60952	1
1975	AL	0.0017%	0.2890	0.0083%	1.4452	0.0417%	7.2261	0.659478
1975	NL	0.0015%	0.2634	0.0075%	1.3172	0.0377%	6.5860	0.633098	1
1976	AL	0.0014%	0.2433	0.0070%	1.2167	0.0350%	6.0837	0.525126	3
1976	NL	0.0015%	0.2592	0.0074%	1.2960	0.0370%	6.4801	0.550841	3
1977	AL	0.0027%	0.5560	0.0137%	2.7799	0.0683%	13.8995	1.265909	2
1977	NL	0.0024%	0.4265	0.0122%	2.1323	0.0609%	10.6616	1.023891	1
1978	AL	0.0019%	0.3894	0.0096%	1.9468	0.0478%	9.7342	0.812532	2
1978	NL	0.0017%	0.3051	0.0087%	1.5254	0.0436%	7.6269	0.60146	1
1979	AL	0.0025%	0.4987	0.0123%	2.4934	0.0614%	12.4670	1.116997	4
1979	NL	0.0021%	0.3679	0.0105%	1.8393	0.0526%	9.1967	0.809232
1980	AL	0.0022%	0.4547	0.0112%	2.2736	0.0558%	11.3678	1.084029	3
1980	NL	0.0019%	0.3262	0.0093%	1.6312	0.0466%	8.1561	0.697493	2
1981	AL	0.0015%	0.2027	0.0075%	1.0133	0.0375%	5.0663	0.417839
1981	NL	0.0016%	0.1890	0.0082%	0.9448	0.0408%	4.7240	0.40454
1982	AL	0.0023%	0.4671	0.0114%	2.3357	0.0572%	11.6783	1.070656	1
1982	NL	0.0016%	0.2847	0.0081%	1.4233	0.0407%	7.1165	0.618683
1983	AL	0.0024%	0.4880	0.0119%	2.4399	0.0597%	12.1994	1.081282
1983	NL	0.0018%	0.3192	0.0091%	1.5958	0.0455%	7.9790	0.66268
1984	AL	0.0022%	0.4572	0.0112%	2.2862	0.0560%	11.4308	1.038968	3
1984	NL	0.0016%	0.2802	0.0080%	1.4012	0.0401%	7.0061	0.583608	1
1985	AL	0.0024%	0.4991	0.0122%	2.4954	0.0612%	12.4769	1.104432	2
1985	NL	0.0017%	0.2980	0.0085%	1.4900	0.0426%	7.4500	0.634401	1
1986	AL	0.0022%	0.4535	0.0111%	2.2676	0.0555%	11.3380	1.030861	2
1986	NL	0.0017%	0.2891	0.0083%	1.4456	0.0414%	7.2281	0.639051
1987	AL	0.0025%	0.5184	0.0127%	2.5921	0.0635%	12.9604	1.246944
1987	NL	0.0023%	0.4014	0.0115%	2.0072	0.0574%	10.0361	0.929027	4
1988	AL	0.0018%	0.3597	0.0088%	1.7984	0.0442%	8.9921	0.773295	2
1988	NL	0.0015%	0.2566	0.0074%	1.2828	0.0368%	6.4138	0.512705	1
1989	AL	0.0017%	0.3436	0.0084%	1.7180	0.0421%	8.5900	0.732705	1
1989	NL	0.0015%	0.2659	0.0076%	1.3296	0.0380%	6.6480	0.551749	3
1990	AL	0.0018%	0.3651	0.0090%	1.8255	0.0448%	9.1276	0.789752	1
1990	NL	0.0018%	0.3080	0.0088%	1.5402	0.0440%	7.7012	0.659922
1991	AL	0.0019%	0.3876	0.0095%	1.9379	0.0475%	9.6895	0.923327	2
1991	NL	0.0017%	0.2990	0.0086%	1.4948	0.0428%	7.4738	0.607063	2
1992	AL	0.0015%	0.3022	0.0074%	1.5110	0.0370%	7.5548	0.688516
1992	NL	0.0017%	0.2907	0.0083%	1.4533	0.0415%	7.2666	0.595099	1
1993	AL	0.0020%	0.4072	0.0100%	2.0360	0.0499%	10.1802	0.998414	2
1993	NL	0.0022%	0.4434	0.0109%	2.2169	0.0543%	11.0845	1.013179	1
1994	AL	0.0027%	0.3816	0.0133%	1.9079	0.0665%	9.5395	1.068219	1
1994	NL	0.0026%	0.3794	0.0131%	1.8970	0.0656%	9.4849	0.918669
1995	AL	0.0025%	0.4493	0.0124%	2.2464	0.0618%	11.2320	1.209444	1
1995	NL	0.0022%	0.3995	0.0110%	1.9974	0.0551%	9.9870	0.985171	2
1996	AL	0.0026%	0.5335	0.0131%	2.6676	0.0654%	13.3382	1.599461	1
1996	NL	0.0021%	0.4249	0.0104%	2.1243	0.0520%	10.6213	1.035637	2
1997	AL	0.0023%	0.4638	0.0114%	2.3188	0.0569%	11.5938	1.224383	1
1997	NL	0.0023%	0.4734	0.0116%	2.3670	0.0580%	11.8351	1.138211	1
1998	AL	0.0024%	0.4957	0.0121%	2.4783	0.0607%	12.3915	1.288368	1
1998	NL	0.0021%	0.4952	0.0106%	2.4759	0.0530%	12.3797	1.198418	2
1999	AL	0.0026%	0.5199	0.0128%	2.5994	0.0638%	12.9969	1.396539	1
1999	NL	0.0025%	0.5775	0.0124%	2.8874	0.0619%	14.4368	1.566046	2
2000	AL	0.0026%	0.5366	0.0132%	2.6829	0.0658%	13.4147	1.517764	2
2000	NL	0.0026%	0.6089	0.0130%	3.0445	0.0652%	15.2224	1.639377	3
2001	AL	0.0026%	0.5297	0.0130%	2.6485	0.0649%	13.2427	1.300609	4
2001	NL	0.0024%	0.5696	0.0122%	2.8478	0.0610%	14.2392	1.318759	1
2002	AL	0.0025%	0.5108	0.0125%	2.5541	0.0627%	12.7706	1.221431
2002	NL	0.0021%	0.4930	0.0106%	2.4649	0.0529%	12.3244	1.15597	2
Total		0.0022%	59.6778	0.0110%	298.3888	0.0552%	1491.9440	142.3178	235
Max		0.0052%	0.6089	0.0261%	3.0445	0.1306%	15.2224	1.9666

First, you'll notice that the odds are best in the Thirties but that the expected values were highest in the last decade or so. That's because of all of the extra teams, and therefore games, that are playing today. More games mean more possibilities to hit for the cycle.

Also, you'll note that the actual cycle totals more closely match 5-plate-appearance expectations that 4-PA or the Avg. expected. There could be a few reasons for this. First that the players who hit for the cycle have better odds to hit a single, double, triple, or home run. Second that the odds to get a hit increase after a player gets a hit, i.e., the theory of the hot bat. Third that players who hit for the cycle do so in high-scoring and therefore high-plate-appearance games, thereby bettering their odds. And lastly, players that are hitting for the cycle are able to better their odds by focusing on their goals.

An interesting study could be conducted by studying the batting records for players who are one hit away from the cycle. Do they raise their own odds? Or is it just the luck that comes from playing in a high-scoring game?

My personal opinion is that there is an element of luck but that players can be streaky and can help themselves focus and achieve a goal such as hitting for the cycle. The higher-than-expected actual totals throughout history indicate to me that something more than dumb luck and a couple extra at-bats in an odd game are at work.