# 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.