No-Hitter Probabilities: What Are the Odds?

A no-hitter turns a pitcher into an instant celebrity in the baseball world. Regardless of what he did before or what he does after, he’ll always be a member of an elite group, a fraternal brotherhood that links Cy Young and Nolan Ryan with the likes of Jose Jimenez, Mike Warren, and Bobo Holloman.

What are the odds of a pitcher throwing a no-hitter? How can we measure a pitcher’s potential for making the leap to celebrity? Who are the top pitchers who have never thrown a no-hitter? Of the pitchers who have thrown no-hitters, who was the least likely? These questions can be answered with some basic probability calculations along with the help of a desktop computer and a thorough statistics database.

When considering the probability of a pitcher throwing a no-hitter we are, in mathematical terms, trying to calculate the probability of a pitcher achieving 27 consecutive outs before allowing a hit. Walks, errors, hit batsmen, and other intervening events don’t concern us as long as there are 27 outs recorded before a single hit is allowed.

Accordingly, there are only two statistics that we need for each pitcher we wish to analyze: outs achieved (innings pitched multiplied by three) and hits allowed. The ratio of outs achieved (OA) to the sum of outs achieved and hits allowed (H) yields the probability of a pitcher getting an out instead of giving up a hit for all the batters he faced that did one or the other, or P(O). Mathematically, we would say: P(O) = OA/(OA+H).

Statistically speaking, the probability of a sequence of independent events is equal to the product of their individual probabilities. So, the probability of achieving 27 outs without a hit is P(O) x P(O) x P(O) . . . x P(O) (27 times). Mathematically, P(NH) = P(O) ^ 27.

I applied this formula to the career statistics for every pitcher in major league history with at least one career start through the 2004 season to find the most likely pitcher to throw a no- hitter. I expected to see a lot of familiar names at the top of the list, but instead I found some statistical anomalies. There are 11 pitchers with at least one major league start who either never allowed a single hit, never got anyone out, or both. If a pitcher never allowed a hit, then the formula above predicts that they have a 100% chance of throwing a no-hitter every time they start a game. If they never achieved an out, then the formula will predict that they have 0% chance of ever throwing one. If, as three of them did, they didn’t give up any hits or achieved any outs, then the formula fails when you try to divide by zero. Fortunately, the pitching careers of these eleven players are not statistically significant—they each had exactly one career start—and we can discount them while we examine the rest of the pitchers. The 11 pitchers are:

 

Table 1. One start and 0 innings pitched and/or 0 hits allowed

Name	IP	H
Dave Pierson (1876 Reds)	0	2
Jay Parker (1899 Pirates)	0	0
Frank DuPee (Chicago WS – 1901)	0	0
Charlie Fritz (1907 A’s)	2q	0
Pat McGehee (1912 Tigers)	0	1
Jim Scoggins (Chicago WS – 1913)	0	0
Dan “Babe” Sherman (1914 Whales)	3	0
Clay Roe (Washington – 1923)	1q	0
Joe Brown (Chicago WS – 1927)	0	2
Marty Walker (1928 Phillies)	0	2
Cesar Tovar (1968 Twins)	1	0

 

After discounting the 11 pitchers above, the one that moves to the top of the list is Randy Hennis of the 1990 Houston Astros. In three career games (one start) he allowed one hit in 9q innings. With 29 outs achieved and only one hit allowed his P(O)=0.9667 and his P(NH)=0.4004, or about 1 in 2.50. Since Hennis’s career of 9q IP is not a good statistical sample, I changed the criteria to only consider pitchers with at least 100 career starts. Table 2 details the 25 pitchers most likely (among the 1,202 pitchers through 2004) with at least 100 career starts.

You may be puzzled by the asterisk by Pedro Martinez’s name. For determining whether or not a pitcher has thrown a major league no-hitter I slightly altered the criteria from those recognized by Major League Baseball. I considered any pitcher who completed nine innings without allowing a hit including those that went on to give up hits in extra innings, such as Harvey Haddix, or were relieved sometime after nine innings.

Pedro Martinez threw nine perfect innings against San Diego on June 3, 1995, while pitching for the Montreal Expos, but gave up a leadoff double to Bip Roberts in the 10th inning. I also included Ernie Shore, who achieved 27 outs (though the first one was already on base) without allowing a hit in relief of Babe Ruth on June 23, 1917.

 

Table 2. 25 pitchers most likely w/at least 100 starts (through 2004 season)
(* has thrown a no-hitter)

Name	IP	G	GS	CG	H	O+H	P(O)	P(NH)	“1 in”
Herb Score	858.1	150	127	47	609	3184	0.808731	0.003241	309
Nolan Ryan*	5386.0	807	773	222	3923	20081	0.804641	0.002827	354
Sandy Koufax*	2324.1	397	314	137	1754	8727	0.799015	0.002339	428
Pedro Martinez*	2296.0	388	321	42	1746	8634	0.797776	0.002243	446
Sid Fernandez	1866.2	307	300	25	1421	7021	0.797607	0.002230	448
J.R. Richard	1606.0	238	221	76	1227	6045	0.797022	0.002186	457
Andy Messersmith	2230.1	344	295	98	1719	8410	0.795600	0.002083	480
Kerry Wood	1043.0	164	164	11	804	3933	0.795576	0.002082	480
Randy Johnson*	3368.0	489	479	92	2612	12716	0.794589	0.002013	497
Sam McDowell	2492.1	425	346	103	1948	9425	0.793316	0.001928	519
Ed Walsh*	2964.1	430	315	250	2346	11239	0.791263	0.001797	556
Joe Wood*	1436.1	225	158	121	1138	5447	0.791078	0.001786	560
Babe Ruth	1221.1	163	148	107	974	4638	0.789996	0.001721	581
Bob Turley	1712.2	310	237	78	1366	6504	0.789975	0.001720	581
Orval Overall	1535.1	218	182	133	1232	5838	0.788969	0.001662	602
Jeff Tesreau*	1679.0	247	206	123	1350	6387	0.788633	0.001643	609
Ed Reulbach	2632.1	399	300	201	2117	10014	0.788596	0.001641	609
Dave Boswell	1065.1	205	151	37	858	4054	0.788357	0.001628	614
Mario Soto	1730.1	297	224	72	1395	6586	0.788187	0.001618	618
Jimmy Dygert	986.0	175	104	62	798	3756	0.787540	0.001583	632
Addie Joss*	2327.0	286	260	234	1888	8869	0.787124	0.001560	641
Jack Pfiester	1067.1	149	128	75	869	4071	0.786539	0.001529	654
Cy Morgan	1445.1	210	172	107	1180	5516	0.786077	0.001505	664
Jose DeLeon	1897.1	415	264	21	1556	7248	0.785320	0.001466	682
Jim Maloney*	1849.0	302	262	74	1518	7065	0.785138	0.001457	686

 

Table 3. 25 pitchers most likely w/at least 100 starts who have never thrown a no-hitter

Name	IP	G	GS	CG	H	O+H	P(O)	P(NH)	“1 in”
Herb Score	858.1	150	127	47	609	3184	0.808731	0.003241	309
Sid Fernandez	1866.2	307	300	25	1421	7021	0.797607	0.002230	448
J.R. Richard	1606.0	238	221	76	1227	6045	0.797022	0.002186	457
Andy Messersmith	2230.1	344	295	98	1719	8410	0.795600	0.002083	480
Kerry Wood	1043.0	164	164	11	804	3933	0.795576	0.002082	480
Sam McDowell	2492.1	425	346	103	1948	9425	0.793316	0.001928	519
Babe Ruth	1221.1	163	148	107	974	4638	0.789996	0.001721	581
Bob Turley	1712.2	310	237	78	1366	6504	0.789975	0.001720	581
Orval Overall	1535.1	218	182	133	1232	5838	0.788969	0.001662	602
Ed Reulbach	2632.1	399	300	201	2117	10014	0.788596	0.001641	609
Dave Boswell	1065.1	205	151	37	858	4054	0.788357	0.001628	614
Mario Soto	1730.1	297	224	72	1395	6586	0.788187	0.001618	618
Jimmy Dygert	986.0	175	104	62	798	3756	0.787540	0.001583	632
Jack Pfiester	1067.1	149	128	75	869	4071	0.786539	0.001529	654
Cy Morgan	1445.1	210	172	107	1180	5516	0.786077	0.001505	664
Jose DeLeon	1897.1	415	264	21	1556	7248	0.785320	0.001466	682
Pete Richert	1165.2	429	122	22	959	4456	0.784785	0.001440	695
Rube Waddell	2961.1	407	340	261	2460	11344	0.783145	0.001361	735
Jake Weimer	1472.2	191	180	143	1227	5645	0.782640	0.001337	748
Tommy Byrne	1362.0	281	170	65	1138	5224	0.782159	0.001315	760
Fred Beebe	1294.1	202	153	93	1090	4973	0.780816	0.001256	796
Doc Scanlan	1252.0	181	149	102	1061	4817	0.779738	0.001210	827
Barry Zito	981.0	153	153	9	832	3775	0.779603	0.001204	831
Gary Gentry	902.2	157	138	25	770	3478	0.778608	0.001163	860
Chuck Estrada	764.1	146	105	24	652	2945	0.778608	0.001163	860

 

Herb Score’s impressive career statistics put him at the top of the list ahead of such notable no-hit artists as Nolan Ryan and Sandy Koufax, though he never actually threw a no-hitter. Table 3 details the 25 most likely pitchers with at least 100 career starts who have never thrown a no-hitter.

The P(NH) in the above tables is the calculated probability that a pitcher would throw a no-hitter in any single start based on his career stats. There is another way to analyze the data using each pitcher’s P(NH) and his number of career starts. With these two numbers it is possible to calculate the probability that a pitcher would throw at least one no-hitter in his career number of starts.

The math is similar to that we used to find P(NH). The prob- ability of a pitcher throwing at least one no-hitter in “x” starts is equal to 1 minus the probability that he would go “x” starts without any no-hitters at all. The probability of a pitcher not throwing a no-hitter in a single start is: 1 – P(NH), or P’(NH). So, the probability of a pitcher not throwing any no-hitters in “x” consecutive games is P’(NH) x P’(NH) x . . . x P’(NH) (“x” times). If we use a pitcher’s number of career starts for “x” we get a product equal to the probability that the pitcher would throw exactly zero no-hitters in his career. Subtracting that probability from 1 yields the probability that the pitcher would throw at least one no-hitter in his career. In math terms, if GS is a pitcher’s number of career starts and P’(NH) = 1-P(NH) is the calculated probability that he would not throw a no-hitter in a start, then the probability that he would throw at least one no- hitter is his career is: P(at least one career NH) = 1 – (P’(NH) ^ GS).

 

Table 4. Pitchers most likely to have at least one no-hitter
(* has thrown a no-hitter)

Name	GS	P(NH)	Probability
Nolan Ryan*	773	0.002827	88.79%
Randy Johnson*	479	0.002013	61.91%
Walter Johnson*	666	0.001361	59.63%
Tom Seaver*	647	0.001365	58.67%
Sandy Koufax*	314	0.002339	52.06%
Roger Clemens	639	0.001139	51.73%
Pedro Martinez*	321	0.002243	51.36%
Don Sutton	756	0.000910	49.75%
Sid Fernandez	300	0.002230	48.82%
Sam McDowell	346	0.001928	48.71%
Jim Palmer*	521	0.001202	46.57%
Andy Messersmith	295	0.002083	45.95%
Steve Carlton	709	0.000865	45.87%
Bob Gibson*	482	0.001238	44.95%
Tim Keefe	594	0.000967	43.72%
Ed Walsh*	315	0.001797	43.26%
Bob Feller*	484	0.001149	42.69%
Catfish Hunter*	476	0.001127	41.53%
Christy Mathewson*	551	0.000948	40.71%
Eddie Plank	529	0.000961	39.86%
Gaylord Perry*	690	0.000716	38.99%
Ed Reulbach	300	0.001641	38.90%
J.R. Richard	221	0.002186	38.35%
Warren Spahn*	665	0.000725	38.26%
Phil Niekro*	716	0.000666	37.96%

 

Not surprisingly, Nolan Ryan is at the top of this list because of his relatively high probability of throwing a no-hitter in a single start and a large number of starts. Based on this analysis it would have been incredibly unlikely that he could go through his career without throwing a no-hitter at all.

In fact, Ryan and five of the other six pitchers with a greater than 50% expectation of a career no-hitter (Randy Johnson, Walter Johnson, Tom Seaver, Pedro Martinez, and Sandy Koufax) have all actually thrown no-hitters. Of the top 25 listed above, 15 (60%) have thrown major league no-hitters. Though he has the highest probability among pitchers with at least 100 career starts of throwing a no-hitter in an individual start, Herb Score doesn’t appear in the this top-25 list because of his comparatively low number of career starts (127).

Through the 2004 season, Roger Clemens tops the list of those with the greatest career probability of a no-hitter who have never done it:

 

Table 5. Pitchers most likely to throw at least one no-hitter

Name	GS	P(NH)	Probability
Roger Clemens	639	0.001139	51.73%
Don Sutton	756	0.000910	49.75%
Sid Fernandez	300	0.002230	48.82%
Sam McDowell	346	0.001928	48.71%
Andy Messersmith	295	0.002083	45.95%
Steve Carlton	709	0.000865	45.87%
Tim Keefe	594	0.000967	43.72%
Eddie Plank	529	0.000961	39.86%
Ed Reulbach	300	0.001641	38.90%
J.R. Richard	221	0.002186	38.35%
Charlie Hough	440	0.001080	37.83%
Rube Waddell	340	0.001361	37.06%
Luis Tiant	484	0.000950	36.87%
Whitey Ford	438	0.001015	35.91%
Greg Maddux	604	0.000735	35.86%
Fergie Jenkins	594	0.000729	35.15%
Herb Score	127	0.003241	33.79%
Bob Turley	237	0.001720	33.50%
John Smoltz	361	0.001093	32.61%
Don Drysdale	465	0.000847	32.56%
Early Wynn	612	0.000636	32.24%
Jose DeLeon	264	0.001466	32.12%
Pete Alexander	599	0.000646	32.08%
Mordecai Brown	332	0.001158	31.94%
Mario Soto	224	0.001618	30.42%

 

Of the pitchers on this list, only Roger Clemens, Greg Maddux, and John Smoltz are active and still have an opportunity to throw a major league no-hitter.

Among the pitchers who have thrown major league no-hitters, who overcame the greatest odds to do so? Table 6 lists the 25 most unlikely pitchers since 1900 to have thrown no- hitters based on the probability of throwing at least one no- hitter in their career number of starts:

Table 6. Pitchers to have thrown a no-hitter based on career probability of at least one

Name	GS	Prob.	Year	Team	LG
Bobo Holloman	10	0.29%	1953	STL	AL
Bud Smith	24	0.52%	2001	STL	NL
George Davis	22	0.81%	1914	BOS	NL
Jose Jimenez	38	0.84%	1999	STL	NL
Mike Warren	27	1.05%	1983	OAK	– AL
Bill McCahan	40	1.45%	1947	PHI	AL
Charlie Robertson	142	2.30%	1922	CHI	AL
George Culver	57	2.31%	1968	CIN	NL
Ed Lafitte	75	2.51%	1914	BKN	FL
Paul Dean	87	2.64%	1934	STL	NL
Ernie Koob	55	2.70%	1917	STL	AL
Juan Nieves	81	2.70%	1987	MIL	AL
Bob Keegan	87	2.85%	1957	CHI	AL
Mal Eason	113	2.86%	1906	BKN	– NL
Clyde Shoun	85	2.91%	1944	CIN	NL
Bobby Burke	88	3.47%	1931	WAS	– AL
Ed Head	53	3.47%	1946	BKN	– NL
Cliff Chambers	113	3.84%	1951	PIT	NL
Nixey Callahan	177	4.01%	1902	CHI	AL
Big Jeff Pfeffer	69	4.05%	1907	BOS	NL
Don Black	113	4.45%	1947	CLE	AL
Don Nottebart	89	4.46%	1963	HOU	– NL
Dick Fowler	170	5.05%	1945	PHI	AL
Derek Lowe	120	5.17%	2002	BOS	AL
Alex Main	44	5.17%	1915	KC	FL

 

Bobo Holloman overcame incredibly long odds to throw his no-hitter considering he started only 10 games in his one- year career with the St. Louis Browns. Derek Lowe and Jose Jimenez are still active and could increase their career odds by accumulating more career starts before they are done, though Jimenez is now a reliever and has started only seven of the 296 games he has pitched in since his no-hitter season of 1999.

Who should you watch today if you want to see a no- hitter? Among the 182 pitchers active in 2004 with at least 50 career starts, the 25 listed below have the highest probability of throwing a no-hitter in an individual start. So, if you want to maximize your odds of seeing a no-hitter, buy a ticket to see Pedro Martinez pitch against Kerry Wood. Some of these pitchers have taken their skills from the rotation to the bullpen (Mike Remlinger, Arthur Rhodes, John Smoltz, and Tom Gordon), but still have the minimum 50 career starts to appear on the list. If they get pressed into starting duty it may be worth seeing—history could be made.

 

Table 7. 25 active pitchers most likely to throw a no-hitter in a single start (min. 50 career starts)

Name	IP	G	GS	CG	H	O+H	P(O)	P(NH)	“1 in”
Pedro Martinez	2296.0	388	321	42	1746	8634	0.797776	0.002243	446
Kerry Wood	1043.0	164	164	11	804	3933	0.795576	0.002082	480
Randy Johnson	3368.0	489	479	92	2612	12716	0.794589	0.002013	497
A.J. Burnett	644.2	102	99	10	535	2469	0.783313	0.001369	731
Johan Santana	624.1	151	75	1	519	2392	0.783027	0.001355	738
Oliver Perez	412.2	70	69	2	345	1583	0.782059	0.001311	763
Barry Zito	981.0	153	153	9	832	3775	0.779603	0.001204	831
Roger Clemens	4493.0	640	639	117	3846	17325	0.778009	0.001139	878
Brandon Webb	388.2	64	63	2	334	1500	0.777333	0.001113	899
John Smoltz	2699.2	602	361	47	2327	10426	0.776808	0.001093	915
Carlos Zambrano	539.2	101	80	4	467	2086	0.776127	0.001067	937
Mike Remlinger	817.0	560	59	4	711	3162	0.775142	0.001031	970
Hideo Nomo	1871.1	301	299	16	1631	7245	0.774879	0.001022	979
Josh Beckett	430.1	77	74	1	376	1667	0.774445	0.001006	994
Mark Prior	446.2	70	70	4	393	1733	0.773226	0.000964	1037
Arthur Rhodes	922.0	551	61	5	816	3582	0.772194	0.000930	1075
Curt Schilling	2812.2	482	370	82	2492	10930	0.772004	0.000924	1082
Tony Armas Jr.	565.0	102	102	0	502	2197	0.771507	0.000908	1101
Al Leiter	2248.2	386	356	16	1998	8744	0.771500	0.000908	1101
Chan Ho Park	1454.2	269	224	9	1294	5658	0.771297	0.000902	1109
Kazuhisa Ishii	473.0	86	86	2	421	1840	0.771196	0.000898	1113
Orlando Hernandez	876.1	139	136	8	780	3409	0.771194	0.000898	1113
Victor Zambrano	495.2	138	64	1	442	1929	0.770866	0.000888	1126
Tom Gordon	1896.2	671	203	18	1703	7393	0.769647	0.000851	1175
Wade Miller	768.0	127	123	5	699	3003	0.767233	0.000782	1279

 

Who is at the other end of the spectrum among pitchers active in 2004? These are the pitchers to watch if you’re an offensive-minded fan and really want to see some hits instead of a pitching duel.

 

Table 8. 25 active pitchers most unlikely to throw a no-hitter in a single start (min. 50 career starts)

Name	IP	G	GS	CG	H	O+H	P(O)	P(NH)	“1 in”
Nate Cornejo	313.0	56	56	3	404	1343	0.699181	0.000064	15707
Josh Towers	348.1	64	52	2	422	1467	0.712338	0.000105	9495
Chad Durbin	331.1	75	56	3	396	1390	0.715108	0.000117	8550
Kevin Jarvis	749.1	174	114	4	894	3142	0.715468	0.000119	8435
Brian Meadows	816.1	218	122	3	972	3421	0.715873	0.000120	8307
Ryan Drese	427.2	80	71	3	502	1785	0.718768	0.000134	7449
Mark Hendrickson	378.1	78	64	3	443	1578	0.719265	0.000137	7311
Jimmy Anderson	574.2	122	96	3	672	2396	0.719533	0.000138	7238
Glendon Rusch	1163.1	225	183	10	1342	4832	0.722268	0.000153	6533
Jason Jennings	607.0	104	104	2	696	2517	0.723480	0.000160	6244
La Troy Hawkins	901.0	443	98	2	1028	3731	0.724471	0.000166	6018
John Halama	816.2	205	114	2	931	3381	0.724638	0.000167	5980
Paul Quantrill	1186.2	791	64	1	1349	4909	0.725199	0.000171	5857
Mike Maroth	539.0	87	87	3	611	2228	0.725763	0.000174	5735
Esteban Loaiza	1663.0	300	263	12	1883	6872	0.725990	0.000176	5687
Jimmy Haynes	1200.2	227	203	2	1358	4960	0.726210	0.000177	5641
Darren Oliver	1407.0	306	228	11	1591	5812	0.726256	0.000178	5631
Elmer Dessens	914.0	249	128	2	1031	3773	0.726743	0.000181	5530
Tomokazu Ohka	665.2	124	115	4	750	2747	0.726975	0.000182	5482
Brian Anderson	1516.1	285	239	12	1704	6253	0.727491	0.000186	5378
Matt Kinney	377.2	98	57	1	424	1557	0.727681	0.000187	5341
Tanyon Sturtze	706.0	187	83	4	792	2910	0.727835	0.000188	5310
Ramiro Mendoza	796.0	341	62	2	889	3277	0.728715	0.000195	5140
Jose Lima	1381.2	312	199	8	1539	5684	0.729240	0.000198	5041
Todd Ritchie	835.2	184	120	7	929	3436	0.729627	0.000201	4969

None of the pitchers on the list above have thrown a major league no-hitter. The most unlikely among pitchers active in 2004 to have actually accomplished the no-hit feat are Terry Mullholland (1 in 4,061) and Scott Erickson (1 in 4,440).

An extra look at Nolan Ryan: Whenever the topic is no-hitters, Nolan Ryan warrants special attention. Since it’s clear he should have thrown at least one no-hitter—nearly an 89% probability—I was wondering how unlikely it was that even Nolan Ryan would do it seven times. Calculating the probability that a pitcher would throw a specific number of no-hitters is only slightly more complex than the math we’ve already done.

Given the probability of throwing a no-hitter – P(NH) – and the probability of not throwing a no-hitter – P’(NH) – and the number of starts to be considered we can use what’s known as binomial probability to calculate the probability of throwing “n” no-hitters in “x” starts.

According to binomial probability, the probability of “n” successes (no-hitters) in “x” trials (starts) is equal to: [P(success)n * P(failure)(x-n) * C(x,n)], where C(x,n) is the number of combinations of “x” items taken “n” at a time. The C(x,n) factor is included to account for the number of different ways to arrange the “n” no-hitters among the “x” starts and is calculated as: (x!)/[(n!)(x-n!)], where x! = 1*2*3* . . . x. When all the pieces are put together it looks far more daunting than it actually is:

P(“n” no-hitters in “x” starts) = [P(NH)n * P’(NH)(x-n) * (x!)/[(n!)(x-n!)]

In Nolan Ryan’s case, P(NH) = 0.0028266, P’(NH) = 0.9971734, and x=773 career starts. The table below shows the probabilities of Ryan throwing 0-7 no-hitters in his career.

 

Table 9: Probability of Nolan Ryan throwing seven no-hitters

No-Hitters	Probability in 773 starts
0	11.213%
1	24.570%
2	26.884%
3	19.585%
4	10.687%
5	4.659%
6	1.690%
7	0.525%

 

You can see that Ryan was far more likely to throw one or two no-hitters in his career than to not throw any at all. In fact, he was almost as likely to throw four (10.687%) as he was to throw none (11.213%)! But even for him, seven no-hitters was quite a feat.

As long as the odds are against them, no-hitters will continue to fascinate fans and immortalize the pitchers that throw them.

 

Acknowledgments

Special thanks to SABR member Pete Hepopski for helping refine my original formulas into the simpler, more complete versions used above following a regional meeting in Minneapolis. Sean Lahman’s database (available at www.baseball1.com) provided all the statistical data.