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August 18, 2005

Crooked Numbers

Royal Flush

by James Click

There's bad, there's the Colorado Rockies, and then there's the Kansas City Royals. If you're into the Jayson Stark "Useless Info" columns, you could easily notch thousands of words about how bad the Royals have been for the past decade or more, a situation only highlighted by their recent losing streak. It's a tough time to be a Royals fan and if you're one of the few, the proud, read on and perhaps you'll feel a little bit better about your team.

To start, let's get some perspective. The Royals' streak of 18 straight losses is not the worst run of baseball of all time. The worst losing streak in the major leagues since 1901 was the 1961 Phillies who managed to lose 23 games in a row from 7/29/61 to 8/20/61. Interestingly, it could have been a lot worse; the Phillies lost five in a row just before the streak, so they actually lost 28 of 29 games in what may very well be the worst month any team has ever had. Here are the rest of the worst:

Year    Team                    Games
1961    Philadelphia Phillies   23
1988    Baltimore Orioles       21
1969    Montreal Expos          20
1943    Philadelphia A's        20
1916    Philadelphia A's        20
1906    Boston Red Sox          20
1975    Detroit Tigers          19
1914    Cincinnati Reds         19
1906    Boston Braves           19

The Royals are on the cusp of greatness, but they're not quite there yet. But how bad is this streak? People have a tendency to grasp onto streaks because they're easily quantified. A team that's lost 15 games in a row is clearly worse than a team that's lost 12 or 10 games in a row. But streaks are as easily broken as they are quantified. Take baseball's greatest streak of all time: Joe DiMaggio's 56-game hitting streak. As many of you know, after his streak was broken DiMaggio hit in another 16 games straight meaning the Yankee Clipper notched a hit in 72 of 73 games. While it's easy to say that DiMaggio's 56-game hit streak was more impressive and improbable than Pete Rose's 44-game streak, but what's more impressive: hitting in 56 games in a row or hitting in 72 of 73 games?

To determine this, we need to get into some binomial distributions. If we assume that DiMaggio had a "true" probability of getting a hit in a game, then the question becomes quite simple. However, that's not quite true because we should instead assume he had a probability of getting a hit in an at bat; as such, things becomes much more complicated. Instead, let's get back to teams and winning games. The odds that a team with a winning percentage w will win any x number of games in a row is simply w^x. Conversely, the odds that they will lose any y number of games in a row is (1-w)^y. This is a binomial distribution, but a very simple one.

If, however, the goal is to determine if a team will win at least 2 of 3 games, the formula becomes more complex because more situations meet the standards for success. For example, if the team wins all three games, wins the first two, wins the second two, or wins the first and last games, all three situations must be counted. The odds of the team winning exactly two of the three games--w2 * (1-w)--must be added to the odds that they will win all three--w3. But since there are three ways in which the team can win two out of three games, that result has to be multiplied by three.

However, the key to the puzzle is Pascal's Triangle, a tool that reveals the binomial coefficient by which each result must be multiplied--three, in the case above. Essentially, the triangle shows how many different ways the final counted result can be achieved by different distributions of the binomial choice. There are three different ways the team can win two games and lose one, but only way in which they can win all three. This is also referred to as "x choose y"--essentially, if one is faced with the decision to choose y games out of x total games, how many possible combinations add up to y.