Was DiMaggio’s single-season hit streak the least likely long streak?

Were there other streaks less likely than Joe's.

Yes, but only one.

If all players had equal ability, then Joe DiMaggio's hit streak would be the least likely by far, simply because it was the longest by far. Think of it this way: no other player in history ever had a 46 game hit streak, but DiMaggio had 11 of them! (Games 1-46 of his famous streak, all the way to games 11-56). He's light years beyond anyone else.

But here's the thing: all players do not have equal ability. The odds against my hitting in two consecutive games would be greater than the odds of DiMaggio hitting in 56. I won't be getting a chance, but unequal ability also affects the odds in the real world.

Hit streaks are a factor of two things:

(1) How likely was a player to hit in each game?

(2) How many games did he play?

In turn, factor one has two components

(1a) How many times did he get to the plate? 

(1b) How likely was he to get a hit each time?

Joe DiMaggio was almost uniquely positioned among modern players to succeed in factor one.

(1a) He played for a great offensive team, and batted near the top of the line-up, so he batted frequently. His likelihood would have been even higher if he had batted lead-off, because he would have come to the plate an additional .33 times per game. That doesn't sound like much, but it would have made his streak four times more likely!

(1b) He was a great hitter who didn't walk much, so he had a high likelihood of getting a hit in each at bat. He was far more likely to get a hit in any given appearance than, for example, Ted Williams, because Williams walked a lot, and walks don't count toward a hit streak.

On the other hand, DiMaggio didn't rank very high on factor two in 1941.

(2) He played in only 139 games, meaning that he only had 84 opportunities for a 56 game single-season streak. Any streak begun in game 85 or later would have to end before it could reach 56.

The great achievement of DiMaggio's streak rests on the power of exponents. Sure, he had an 80% likelihood to get a hit in any given game, but any decimal taken to the 56th power becomes a very small decimal. Even with Joe D's great ability, the odds against that hit streak were about 8000-to-1.

DiMaggio's streak was highly unlikely, even with DiMaggio's hot hitting that year. If you were to replay DiMaggio's season again and again on an infinite loop with our statistical model of his 1941 performance, he would reach 56 or more every 7,797 seasons. It was a statistical anomaly, even with expectations raised in deference to the talent of the great DiMag.

But Uggla's streak was beyond anomalous and into the territory of miraculous. Dan Uggla was less likely to hit in 33 consecutive games than DiMaggio was to hit in 56. Far less likely. About 16 times less likely. Somehow, Uggla hit in 33 consecutive games despite having fewer plate appearances per game than DiMaggio and a much lower probability of getting a hit in any one of them. How did it happen? Uggla, a mediocre hitter, simply went on one of the all-time hot streaks. In those 33 games, he batted .377 with 15 homers. During the rest of the year, Uggla batted .194. That 183-point difference is uncanny.

DiMaggio batted .408 during the streak and “only” .321 the rest of the year, but it isn't especially strange for a hitter of DiMaggio's caliber to have a long hot streak in which he maintains a .400 average. In 1939, Joe D was batting .443 for the entire season as late as July 13. The explanation for 1941 is not that Joe D avoided hitless games. He had many of them, 25 to be exact. The magic of 1941 is how those 25 hitless games were distributed. They managed to stay out of the streak. That distribution was rare, but there was really no radical upswing in his performances. It is not unusual for DiMaggio to hit like DiMaggio.

But it is unusual for a .194 hitter to hit like DiMaggio, and Uggla did just that. DiMaggio's OPS was 1.181 during his streak. Uggla's was 1.200 in his.

Nobody can explain it. It just is.

Were there streaks that were predictable?

Yes. Sort of. There was never a streak of 30 or more that was an even money bet, but there were a few that came pretty close,

If the model were to recreate Ed Delahanty's season on that infinite loop I mentioned, it would reveal that Big Ed would post a 30-game streak approximately every five years. While DiMaggio was performing at the top of his game, and Uggla seemed to have signed a short-term pact with Satan, Delahanty was simply conducting business as usual. Yes, he batted .422 during the streak, but he also hit .406 in his other games. That mere 16-point difference compares to 87 for DiMaggio and 183 for Uggla.

Were there individual seasons that were even more streak-friendly, but didn't produce one?

Yes, there is one that really stands out, and a few honorable mentions, but there is nothing in the past hundred years.

Hugh Duffy had the most likely season to produce a 30-game streak. No other season came close. He accumulated nearly five plate appearances per game and got a hit in more than 38% of his plate appearances. Both factors are close to all-time records at the modern pitching distance. He was so efficient in 1894 that he was about an even bet to pull off a 30-game streak, but he did not. He did reach 26.

Of the eight seasons most likely to produce a 30-game hit streak, only three actually did.

Note that George Sisler has two seasons on the list. His 1920 season did not produce a 30-game streak, but his 1922 season more than made up for it with a 41-game streak, which remained the 20th-century record until DiMaggio obliterated it in 1941, and still ranks third in the century behind DiMaggio and Pete Rose.

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FOOTNOTE:

The calculation is season-based, not streak-based. In other words:

* It calculates how often how often the characteristics of a particular season will produce a streak of 56 games or more. In DiMaggio's case, such a streak will occur every 7,797 years

* It does not calculate the frequency of 56-game streaks. One DiMaggio streak will occur on the average, every 1,554 years. This is because some years will include multiple streaks of exactly 56 games. They come in clumps. A 57-game streak, for example, is by definition two 56-game streaks, one from game 1 through game 56 and one from game 2 through game 57. In that sense, as noted at the beginning of the article, DiMaggio has 11 streaks of 46 or more. While this is mathematically correct, it is not especially useful information, since we think of DiMaggio's streak simply as 56 games, not as 11 consecutive 46-game streaks.

Thus, in 100,000 such seasons:

* There will be 64 streaks of exactly 56 games (one per 1554).

BUT

* There will be only 13 seasons including a streak of 56 or longer (one every 7997).