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# Fun With Math: A Breakdown of the 49ers' Playoff Chances

So let's say you're sitting there with visions of sugar plums and Niner division championships dancing in your head. Let's also say you're looking at the NFL schedule for Weeks 15-17 (and checking it twice). Now, if you're with me so far, and you're either a nerd with little time to spare, a jock who used to cheat off of the nerds' exam sheets, or just reeeeeeeeeeeally feeling the Christman spirit, then this post is for you.

For the past couple of weeks, my friends and I have been making half-hearted attempts to figure out the Niners' chances of making the playoffs given various combinations of wins and losses by San Francisco, Seattle, and St. Louis in their remaining games. After reading Fooch's wild, wild West post yesterday, I decided a full-hearted attempt was worth my time, and would benefit those Niners Nation readers who've been pulled back in just as they thought they were out.

So, here's what I did. The 49ers, Seahawks, and Rams each have 3 games left, which means there are 9 total games remaining between the 3 teams. Ordinarily, in the world of probability, these would be considered 9 independent events; in other words, 9 total games that each team can either win or lose (I'm ignoring ties because they happen about once every 1,000 games). And in that same probability world, assigning an equally likely win or loss to each team for each of the 9 games results in a total of 512 possible win-loss combinations (i.e., 2^9 = 512). I'm generous with my time, but not that generous!

Luckily for me -- and I guess, you -- the fact that SF plays STL in Week 16 and STL plays SEA in Week 17 means that 2 of the events are dependent rather than independent; meaning that whether one team wins or loses these games depends on whether or not their opponent wins or loses. In other words, this isn't first grade; SEA and STL don't both get a trophy (read: win) for just showing up to their Week 17 contest. Thankfully, in probability world, this fortunate scheduling situation reduces the number of possible win-loss combinations from 512 to 128 (i.e., 2^7 = 128). Long story short, I took these 128 possibilities (aka scenarios), and figured out which team would win the NFC West in each of them given (a) the current standings situation, and (b) the combination of wins and losses in each scenario.

For example, in one scenario, the Seahawks finish L-L-W, the Rams finish W-L-L, and the 49ers finish W-W-W. In this scenario, the 49ers win the NFC West outright with an 8-8 record, 1 game ahead of both the 7-9 Seahawks and the 7-9 Rams. Awesome! A nice illustration and mathematical proof that the Niners have at least a 1-in-128 chance to make the playoffs!

Actually, after doing this kind of win/loss assignment thing for each of the 128 scenarios, it turns out that the 49ers win the division in 25, or 19.5% of the time. So, assuming each team has an equally likely (aka 50/50) chance to win each of their remaining games, the 49ers would make the playoffs about 20% of the time. Or, to use the always-useful coin-flip analogy, if each of the remaining games were decided by a coin flip -- rather than a 60-minute battle of brains and braun -- the 49ers would make the playoffs about once out of every 5 sets of coin flips. Now, whether or not each game is actually a 50/50 proposition is for your brain to decide. I'll just provide the raw theoretical probabilities, and leave the whimsy up to you. But more on this at the end of the post...

Full results and analysis after the jump...

I tried uploading my Excel table as a picture on here, but the dimensions were all screwy, so below is the link to a .pdf file with results for all 128 scenarios (any specific scenario numbers I mention in this post will be in reference to the table in the .pdf) :

And, for convenience, here's a reproduction of the summary table at the bottom of the .pdf:

The way to read this table is to say, "OK, what's the probability of the Niners winning the division if they finish at 7-9?" Find the "7-9" row in the "Record" column, then go across to the "SF" column, and there's your answer: 22.9%.

Now, while the devil's really in the details of that .pdf, this summary table tells us a couple of things we should understand right off the bat. First, despite my eager anticipation of a collective nervous breakdown on the part of NFL pundits -- not to mention the ensuing ridiculous debate about changing the playoff seeding rules based on 1 incredibly atypical season -- there's absolutely no way the 49ers can win the division at 6-10. In fact, none of the 3 teams can win the division at 6-10. The reason for this is simple: The SEA/STL game in Week 17 ensures a 7-win team in the NFC West (again, ignoring the potential for a tie). Oh well, guess I'll just have to wait for the next marginally violent hit delivered by James Harrison to laugh at hyperventilating NFL pundits.

Also a direct byproduct of the SEA/STL game in Week 17 is that, if either SEA or STL wins out, they're ensured a division championship. That's because the SEA/STL game renders it impossible for both of the teams to end the season tied at 9-7.

OK, so those are the stone-cold locks; the stuff that doesn't take gratuitous amounts of nerdness to figure out. Before moving on to some specific scenarioes that are worthy of mention, let's get a couple of additional (quasi-)truisms out of the way:

1. So we're all on the same page here, the NFL's divisional tiebreakers go -- in order of application -- (a) head-to-head win %, (b) win % against NFC West opponents, (c) win % against common opponents, (d) win % against NFC opponents. If it's a 3-way tie, "head-to-head win %" turns into, "win % in games between the 3 teams."
2. Despite what you might hear from imaginative friends or mistaken pundits, there's absolutely no way for the 49ers to lose any 1st-place tiebreaker with the Seahawks and/or Rams. Keeping in mind that the Niners can only make the playoffs at 8-8 or 7-9, here's the explanation. In order to get to 8-8, the Niners would have to win both of their remaining division games, which would thereby give them a season sweep against STL (i.e., they'd win the first tiebreaker in a 2-way tie with STL), an insurmountable 5-1 division record (i.e., they'd win the first tiebreaker in a 2-way tie with SEA), and an insurmountable 3-1 record in games among the 3 teams (i.e., they'd win the first tiebreaker in a 3-way tie). A 7-9 finish for the Niners is a little more complicated to figure out, but the same basic principle applies: In the 11 scenarios where the 49ers finish tied for 1st place at 7-9 (#s 118-128), there's no mathematically possible way for SEA or STL to beat the 49ers in any of the tiebreakers. In other words, the only way the 49ers don't make the playoffs at 7-9 is if either SEA or STL wins the division outright, i.e., 1 of the other 37 scenarios -- in which the 49ers finish 7-9, but not tied for 1st -- ends up happening.

On to some more-nuanced fun with probabilities. For these, I'm going to go ahead and assume that the 49ers don't win out to finish 8-8. Obviously, we're all rooting for that, but (a) it's not -- in my opinion -- highly rational to think a team that won 4 of their first 12 games would win 4 of their last 4; and (b) it's not all that interesting for the purposes of this post. As the table above showed, the Niners make the playoffs in 14 of the 16 scenarios in which they finish 8-8. The only way they don't make the playoffs at 8-8 is if the Seahawks also win out. And, in terms of that possibility, I might add that thinking SEA will win out against TB, ATL, and STL is -- in my opinion -- equally irrational, especially in light of the drubbing they just took at the hands of the aforementioned 4-wins-through-12-games team.

SPECIFIC SCENARIOS INVOLVING A 7-9 NINER FINISH

So, given the certainty of another playoff-free season if the 49ers finish 5-11 or 6-10, and the near-certainty of a playoff berth (finally!) if they finish 8-8, it should be obvious now that all of the playoff scenario hijinks occur if the Niners finish 7-9. With about 5 seconds of thought, it becomes equally obvious that there's only 3 ways for the Niners to get to 7-9:

1. W @ SD, W @ STL, L vs. ARI
2. L @ SD, W @ STL, W vs. ARI
3. W @ SD, L @ STL, W vs. ARI

Let's take these one by one...

1. Of the 16 possible scenarios in which the 49ers finish W-W-L (#s 6, 14, 17, 24, 31, 38, 45, 73, 76, 79, 82, 118-122), they win the division in 5 (#s 118-122), or 31.3% of the time.
2. Of the 16 possible scenarios in which the 49ers finish L-W-W (#s 7, 15, 18, 25, 32, 39, 46, 74, 77, 80, 83, 124-128), they win the division in 5 (#124-128), or 31.3% of the time.
3. Of the 16 possible scenarios in which the 49ers finish W-L-W (#s 1, 9, 20, 27, 34, 41, 53, 57, 61, 65, 69, 85, 89, 93, 97, 123), they win the division in 1 (#123), or 6.3% of the time.

To understand the moral here, remember that the 49ers have -- overall -- a 22.9% chance to make the playoffs if they finish 7-9. Therefore, given the above paths to 7-9, a win over the Rams in Week 16 increases their chances to 31.3%, whereas a loss to the Rams decreases their chances to 6.3%. In other words, a 7-9 Niner team is 5 times more likely to make the playoffs if they beat STL than if they lose to STL. The basic reason for this is that a loss to the Rams makes the path to 8 or 9 wins a lot easier for STL, which thereby makes winning the division outright a lot easier for STL, and indirectly makes it a lot easier for either SEA or STL to finish at least 1 game ahead of the 49ers. So, the bottom line here is that the Rams game is essentially a must-win for the Niners.

Now, let's turn to an issue that's been a source of debate in some of the 49er-related comment threads around the web: How much does a (likely) loss to SD this week hurt the 49ers' playoff chances. The answer is, "It depends on what the Seahawks and/or Rams do in their Week 15 games." In this little exercise, we're still assuming that the Niners will win 2 of their final 3 games to finish 7-9, so we basically just need to break down the 16 L-W-W scenarios detailed a minute ago:

1. Of the 16 possible scenarios in which the 49ers finish L-W-W, the Seahawks AND Rams lose their Week 15 games in 4 (#s 46, 125, 127-128). Of these 4 scenarios, a 7-9 49er team wins the division in 3 (#s 125, 127-128), or 75.0% of the time.
2. Of the 16 possible scenarios in which the 49ers finish L-W-W, the Seahawks OR Rams lose their Week 15 game in 8 (#s 15, 18, 32, 39, 80, 83, 124, 126). Of these 8 scenarios, a 7-9 49er team wins the division in 2 (#s124-128), or 25.0% of the time.
3. Of the 16 possible scenarios in which the 49eres finish L-W-W, the Seahawks AND Rams win their Week 15 games in 4 (#s 7, 25, 74, 77). Of these 4 scenarios, a 7-9 49er team wins the division in 0, or 0.0% of the time.

This, I think, is where things get incredibly weird. Essentially, the moral here is that in Week 15, the Seahawks and Rams control the 49ers' playoff destiny. If both of them win, the Niners are eliminated from playoff contention. If only one of them wins, then the Niners are still below 50/50 to win the division, even if they were to beat the Rams and Cardinals. However, if neaither SEA nor STL win in Week 15, it then becomes more likely than not that the Niners will win the division if they finish W-W. To be clear, they wouldn't technically control their own destiny in Week 16, per se. Rather, losses by SEA and STL this week would just mean that the only way for the 49ers to not win the division is if SEA also finishes W-W to win the division outright at 8-8.

Let me just sum this section up by looking at things from a rooting-interest standpoint. Obviously, we want the Niners to win out. Short of that, the scenario-based probabilities suggest we want them to beat the Rams in Week 16 and the Cardinals in Week 17. Of course, we obviously want the Niners to beat the Chargers this week too. But, short of that, the scenario-based probabilities suggest we want both the Seahawks and Rams to lose this week. Short of that, we want at least one of them to lose this week. And, finally, what we absolutely do not want to see this week is the Seahawks beating Falcons and the Rams beating the Chiefs. That result would mean a Niner elimination.

QUICK NOTE IN CLOSING

Back near the beginning of the post, I mentioned that you'd be justified in disagreeing with my assumption that each team has a 50/50 shot to win each of their games. I mean, as I've alluded to in the post, the Niners probably aren't even money to beat the Chargers (in fact, Vegas currently has them as about a 3-to-1 underdog). The same could be said about SEA's chances against ATL this week. OK, so let's say I grant you that the scenario-based probabilities I've presented in this post are based on a mindless exercise that's akin to flipping coins, picking numbers from a hat, rolling dice, etc.

But, here's the thing. If you look at a scientific, stat-based estimate of the playoff probabilities for SF, SEA, and STL as they stand at this very moment, you'll find numbers that are incredibly similar to those resulting from my mindless exercise: SEA = 35.2% (vs. my 40.6%), STL = 45.9% (vs. my 39.8%), and SF = 18.8% (vs. my 19.5%). To be sure, there's no doubt that the differences are due to predictive factors like (a) STL's weaker upcoming opponents relative to SEA & SF, (b) Matt Cassel's appendectomy-induced uncertain status this week, (c) SEA's & STL's 1-home-game advantage over SF, (d) SEA having the toughest single opponent, etc. However, the entire weight of NFL statistical research is going against me in this little prediction contest, and my relatively simplistic probabilities are -- at worst -- 5% off?

So, you see, assuming each game is a 50/50 proposition isn't such an off-base assumption here. We're talking about a pretty random division being contested by 3 pretty random and at-best-average teams playing 3 pretty similar schedules. If we were talking about a different division or different teams or different schedules, then a "random" exercise like mine should, in fact, be quickly discarded. But we're not, so it shouldn't be.

BOTTOM LINE

1. Obviously, root for the Niners to win all 3 remaining games. But, short of that...
2. Let's go Falcons and Chiefs this week!
3. Niners, please beat the Rams next week!