## 2015-09-21

### Probabilistic Baseball Analysis

If you put a classical particle in a box with energy E1, and the height of the box is such that the sides have dimension E1+E2, the particle will not be able to escape the box.

If you put a sub-microscopic particle in a box with energy E1, and the height of the box is such that the sides have dimension E1+E2, the particle does have a nonzero chance of "tunneling" through the box and emerging on the other side.

Baseball teams are not sub-microscopic.

So you'd think that a team which has a 0% chance of making the postseason will not possibly make the postseason under any circumstances. This is what hockey fans know well as the dreaded "mathematically eliminated". Major League Baseball likes to use magic numbers, though in the wild card era they've lost a little of their...[obvious pun here deleted. -ed]. (Even without wild cards, however, magic numbers aren't wholly accurate since they don't take the schedule into account).

Possibly excepting the the Houston Astros.

However, if you've heard that your favourite team has a 100% chance to make the postseason, you'd be happy, right? In theory, you could stop watching the games altogether (which probably adds a sinister conspiracy theory to what comes next). So why does the official MLB.com postseason probability calculator have teams falling out of 100.0%?

Above is the screenshot for the NL East playoff probabilities for the morning of September 21st. The Washington Nationals aren't out of the hunt just yet! They have a whopping 1.2% chance of making the postseason. Specifically, they have a 78-71 record with 13 games left to play. If they win all 13, they'll have a 91-71 record. Their opponent, the 84-65 NY Mets, also have 13 games left to play and could go 6-7 to finish with a 90-72 record. Most critically, the Mets and the Nationals play each other for the last three games of the season. Over the next 10 games, Washington needs to "catch" only four wins against NY to enter the series within striking range of the Mets. If the Mets go 3-7 and the Nationals go 7-3 over the next 10 games, they will enter the series with Washington only 2 games back. Sweep the Mets, enter the postseason. It's not impossible, though would be very difficult, and I guess that's what a 1.2% probability gets you.

Mathematics doesn't care much if you're on a hot streak, but in the real world we probably should, if for no other reason than to predict future performance in a world where the trade deadline and injuries exist. What the Mets did in early June is interesting but for our purposes not entirely valid. Juan Uribe is going to probably miss a couple upcoming games with his bruised chest injury, Dario Alvarez is out for two weeks with a groin injury just days after being dubbed the "lefty-specialist the Mets have desperately needed", Jacob deGrom is having his start pushed back (and held to an informal innings limit), and Matt Harvey's innings limit is the talk of the league. On the other side, Washington has picked up Jonathan Papelbon at the trade deadline (replacing Drew Storen looks pretty smart now, doesn't it? Storen is out for the season). Losing Carpenter and Storen and Zimmerman hurts, but still this isn't the same team that played games in July. The Mets may not be as good as their record. Washington may be better than theirs. We don't know the specifics of the analysis Baseball Prospectus used other than it featured the Monte Carlo Method (more on that in a minute), but I assume it doesn't blindly assume the probability of winning each game is 50/50.

Still, take away this important fact from the 1.2% number: the Washington Nationals are not eliminated from the postseason. They can (in a not entirely unrealistic scenario) win two games against the stumbling Orioles (73-76), sweep the embarrassing Phillies (56-94) at home, win the rain-delay forced game against the Reds (63-85), and then depending how the Atlanta series goes, they could be facing a struggling Mets team only a game or two back for the final showdown of the season. The pennant could be decided on Sunday October 4th! (They could end up tied too, forcing a single game playoff on October 5th, but that would be even more exciting.)

If you're a fan of, say, the aforementioned Atlanta Braves, you wish your team had a slim chance to make the postseason. Let's assume that the Mets win their remaining games. That will give them a 97-65 record. If you're an Atlanta Braves fan on the morning of Wednesday August 19th, your team has just lost its 66th game and wouldn't be able to beat it. Our theoretical Nationals Pennant winner will have an 88-74 record, Atlanta lost its 75th game on August 29th (over three weeks ago). Not surprisingly, Atlanta has a 0% chance of making the postseason. Win their remaining 12 games and enjoy that 72-90 record that wouldn't even put them in second place in the NL East today. If your team has a 0% chance of making the postseason, you're done. Got it?

Good. Because it's wrong, apparently.

Here's a screenshot of how things looked a week ago on September 15th. At 8am on September 15th, the Mets were 83-61 following a win over the Miami (Editor: Insert name of Miami team here) [your guess is as good as mine... -ed]. Washington was 73-70 and had just netted its second win in a row after a brutal 5-game losing skid where they narrowly avoided being swept by the lowly Miami (Editor: Insert name of Miami team here), and did get swept by (who else but) the Mets. So the Mets were 10 games up on Washington with 19 games left to play. It didn't look good for Washington, I'll grant you. Hell, it doesn't look good for them now, but they still have a chance to come in first.

The problem is that they had a chance to come in first on September 15th as well. We know this because they have a chance right now, and we had to get here from September 15th somehow: so the "road" these two teams took to get here has to be part of the mathematical analysis done on September 15th. In fact, it isn't the only road that could have led them here. Between September 15th and today, the Nationals have lost only one game (again to Miami, who are seemingly enjoying play spoilers this September), which means that the math has to assume both zero and one loss over the 6-game span. Going on a 5-1 or 6-0 streak isn't that mathematically impossible, especially in a stretch that includes two more games against the Phillies and a 4-game series against Miami (and Washington was playing in their home park). Likewise, the Mets beat the Yankees on Friday, so the Bronx Bombers didn't get the sweep. On the morning of September 15th, the number crunchers surely factored in that the Mets could go 1-4 or even 0-5 over that stretch. Surely.

So why did the Nationals have a 0% probability on that Tuesday morning? If we assume our 88-74 final record for Washington, that means they had to win 15 out of 19 games (0.789 win percentage) while the NY Mets have to win only 4 out of their remaining 18 games (0.222 win percentage). Again, not likely I admit (the Mets remaining schedule was/is pretty easy except for the 3-game subway series against the Yankees), but obviously higher than 0%.

You might argue that this is all just a lazy rounding issue, where 99.9999% turns into 100.0% when formatted for the website. As proof, you'd say to look what happens a day later.

It's now the morning of September 16th. The Mets have moved on to play the anemic Miami (Editor: Insert name of Miami team here) in Shea Stadium First National City Bank of New York Park, and...lost. The Washington Nationals just won their third game in a row, blanking the Phillies 4-0. New York has a 83-62 record while Washington's is 74-70. For our "Washington finishes with a 88-74 record and NY has a 87-75 record" season, the Mets have to have a win percentage of 0.235 and the Nats have to have a win percentage of 0.778. Well, 0.235941174 and 0.777777777, I probably shouldn't round in a discussion of how rounding isn't the issue. According to the POST data on the site, the Mets went from 99.8% to 99.1% over that single day. Doesn't that seem like far too much? Regardless, 99.8% is not 100%. Unfortunately, it all comes clear when you look at one more stats page.

According to the website, the St. Louis Cardinals, the Pittsburgh Pirates, and the Chicago Cubs are all guaranteed to make the postseason. Guaranteed. 100.0%. Now the St. Louis Cardinals have indeed clinched a postseason appearance, being the first team of the 2015 postseason to guarantee a playoff spot despite losing to the Cubs. But what of the other two teams? Have they also clinched the postseason? The quick answer is no. (This is frequently the quick answer when the issue of the Cubs and the postseason is concerned).

The San Fransisco Giants are playing another 13 games, 10 of which are against opponents not named the LA Dodgers. The Washington Nationals are playing another 13 games, 10 of which are against opponents not named the NY Mets. Can both of these teams win 9 or more games to pull ahead of the lovable losers from northside Chicago? Yes. Remember that the Cubs are playing a 3-game series against Pittsburgh. If the Pirates sweep them, they have at minimum 92 wins on the season. That leaves the Mets, Nats, Giants, and Dodgers all in a position where their wins can exceed the current Chicago total. Even the Blue Jays, +9 games up on the LA Angels of Anaheim can't plan the parade yet, their magic number -- well, technically the Yankee's Magic Number -- is 12.

This is a common problem for Toronto sports teams.

Next week you could wake up to check the MLB website and discover that the Chicago Cubs probability of making the postseason has fallen below 50%. Hell, you might wake up next week to find that the Mets are in the same boat. (Though that would be really unlikely). The point is, that it's not 0%. Nobody's at 100% except for St. Louis. The rest of it is just a glitch of rounding or more fundamentally the problem inherent when using Monte Carlo simulations based on Baseball Prospectus' assumptions.

Baseball teams, despite what a dumb reading of the numbers would tell you, don't follow quantum mechanics.