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Probability. That’s what it’s really about though. And the truth is that it’s damn hard to estimate. Much, much harder than you’d think. Even smart people, who are supposed to have expertise in a subject, get it shockingly wrong depending on how you ask them. The human brain just has too many biases – mental shortcuts it likes to take – to make accurate perception of unlikely events something that anyone is innately good at. Read Daniel Kahneman’s seminal book Thinking, Fast and Slow if you’re interested or don’t believe me. But one thing humans are blessed with is the faculty of reason. It’s harder, it generally takes longer, and it’s rarely fun, but it is something we can get better at. When we need to, we can slow down and actually reason out an answer. This is particularly effective if you’ve learned tools that you can apply your reason to. So, is that statement I started with actually reasonable? Just how much of a miracle will it be if Toronto FC, a team that makes losing special, actually pulls off a great escape from further infamy? Well, the first thing to remember is that a (non-knockout) soccer game has three, rather than two, potential outcomes: win, loss, or draw. It seems obvious, but people often forget that last one when they start an ad hoc mental estimate. Draws are important because, unlike most other major team sports, they’re really common in modern, low-scoring soccer. Keep in mind, TFC will be eliminated with the loss or draw of either of their next two games, or if Columbus win or draw either of theirs. So, right from the beginning, we can do a very basic calculation. How many permutations of results involving Toronto FC and the Columbus Crew exist with two rounds left in the season? That’s two games for Toronto, and two games for the Crew. Each one has three possible outcomes. TFC needs the exact correct outcome in every single one, or they’ll miss the playoffs for the eighth consecutive year. (1/3)*(1/3)*(1/3)*(1/3) = 1 / (3^4) = 1/81 There are 81 possible results. TFC makes the playoffs in one of them. 1:81 . That’s it. That “we win two, they lose two” scenario your brain has such an easy time conjuring? There are 80 other possible outcomes you haven’t taken the time to consider. Converted into a percentage it’s about 1.23%. Assume an equal chance of a win, loss, or draw in each match, and TFC has a 1.23% chance of making the playoffs. “But Mike, there isn’t an equal chance of a win, a draw, or a loss in each game! Teams don’t draw one-third of the time!” I hear you, I hear you! And you’re absolutely right. So let’s refine our analysis a little, no? I’ve taken the time to pull some data from MLS’s official site. To date, in 2014, at the time of writing before Portland and RSL kickoff, there have been 84 draws in 305 MLS games this season. That’s about 27.5%. So less than one-third, but actually not all that far off. How does that number change our analysis? Well the formula is ((1-0.275)*5)^4 . In words: (The chance of there not being a draw times the chance the team TFC needs to win does) repeated four times. Remember, any draw, in any of the four relevant matches, eliminates TFC. AND Toronto needs the right team to win in each match. It’s not encouraging, but it’s better than before, and works out to a 1.7% chance of making the playoffs. Hey, that’s almost 1 in 50! Roll on, TFC! Now, the chance of drawing the above is premised on is for the entire league. Maybe we should just look at TFC and Columbus? It turns out that the chance of a draw in one of their games this season is 26.6%. So really, not all that different than the league stat. For all of Chicago’s attempts to redefine what we think possible in terms of drawing, it really hasn’t shifted the overall likelihood very much. That points to a stability in all these numbers that we should store in the back of our head. And it does put TFC’s chances of making the playoffs all the way up to 1.8% using our last formula! Baby steps. “But Mike, there’s not a 50/50 chance that either team will win, even in a non-draw… Surely TFC has a better than 50% of beating the Impact! And home teams always win more than away teams.” Okay, you got me again. I’m going to have to go a step further. Fortunately, the fine folks at ESPNFC.com, when they’re not not correcting silly Jermain Defoe blog pieces, present MLS’s table split for home-and-away. That makes it possible to calculate both team-specific and overall numbers for the chance of: a home win, a home draw, a home loss, an away win, an away draw, and an away loss. As Ben Knight used to write, “onward!” Overall, in MLS in 2014, the home team wins 48.2% of the time. They lose 24.3 % of the time. As above, games end in draws 27.5% of the time. I’ll let you figure out what that means for away teams… Considering that, in sequence, we need 1 home win (TFCvMTL), 1 home win (NYRvCMB), 1 away win (NERvTFC), and then another 1 away win (CMBvPHI) the math is pretty simple. It’s just 0.482 * 0.482 * 0.243 * 0.243 = 0.014 = 1.4%. Oh. Oh, that’s bad again. Maybe it’ll get better if we figure out the team-specific numbers? Here’s Columbus: home win 50%, home draw 25%, home loss 25%, away win 25%, away draw 37.5%, away loss 37.5%. Toronto FC: home win 43.8%, home draw 12.5%, home loss 43.8%, away win 25%, away draw 31.3%, away loss 43.8%. (TFC’s numbers not totalling to 1 is a rounding error. The formulas use more precise values.) Now, we need a TFC home win, a Crew away loss, a TFC road win, and then a Crew home loss. That’s 0.438 * 0.375 * 0.25 * 0.25 = 0.01 = 1%. That’s really bad! A 1:100 shot. Maybe we should have stopped back at the simple analysis after all! It’s worth noting that this is probably close to how sportsclubstats.com is generating their estimate as well. Okay, okay, okay! I can hear it already! Yes, TFC is playing Montreal. Montreal is terrible and hasn’t won a game on the road all season. Heck, they’ve barely drawn. And Columbus has to go to New York who haven’t lost at home in however many straight home games! That does count for something, but how do we model it? A simple way would be to look at each opponent’s relevant team-specific number and average the two values. For instance, Montreal has lost 12 of 16 away games, or 75%. TFC’s chance of a home win (0.438), times Montreal’s chance of an away loss (0.75), over two, or (0.438+0.75)/2 = 0.594 = 59.4% . Call it 60% for clarity. Maybe TFC has a chance after all? The Red Bulls have won 10 of 16 home games so for their match versus Columbus the calculation is (0.625+0.375)/2 = 0.5 = 50%. New England have lost 25% of their home games, the same percentage of away games that TFC have won, however, so that value doesn’t change at all. It’s the same with Philly’s away wins of 25%. That’s the same as the Crew’s home losses, so there’s no change there either. Plugging in the revised “crappy Montreal” and “good New York” numbers does improve TFC’s chances though. 0.60 * 0.50 * 0.25 * 0.25 = 0.019 = 1.9%!!!! There you go folks, TFC has gotten within touching distance of the 1:50 threshold! Now you could go even further and add a subjective component like motivation, or form, and play around with those base numbers. I’ve done it, and even using the most generous estimates possible you’ve got to really stretch credulity to get Toronto even close to a 5% chance of making the playoffs. And the reality is, that’s probably the wrong way to be estimating because the father you pull away from the underlying base rates of the entire league, the less you’re allowing for regression to the mean. When you get right down to it, TFC probably has somewhere between a 1:50 and a 1:100 chance of making the playoffs. That’s not nothing, but it really shows how much of a miracle it’ll be if it happens. You don’t have to stop cheering, or stop hoping – that is the big virtue of how deceptively simple “we win two, they lose two” is – but just don’t be too surprised if it doesn’t work out. I mean, it is TFC we’re talking about. Oh yeah, and Defoe’s “injured”, Bradley’s suspended, and the rumors are TFC is trying to add a big pile of stupid to their already existing collection. Fun times. Correction (14/10/18, 11:15 a.m. ET): An earlier version of this article incorrectly substituted the chance of a Columbus home win for the correct chance of a Columbus away loss in calculating the team-specific averaged base rate of Columbus not losing in New York. The correct value has now been substituted, the calculation adjusted, and subsequent paragraphs modified. It turns out that TFC has an even worse chance of making the playoffs than the original article estimated. Yippee!