Picking Playoff Series Length

People who are good with numbers may have noticed that I picked an unreasonably large number of first round series' to be 5 games long. In fact, people who aren't good with numbers probably noticed it also because I picked 7 out of 8 to be the same length. It turns out however, that for a series with a clear favorite, that's the most likely out come. In fact, there are easy rules to follow.

*This all assumes that home teams win 60% of the time with equal skill. This also does not apply to the finals which have a different format (2-3-2).

When to...

Pick the home team in four

As I noted last year, sweeps are surprisingly unlikely. It's even more unlikely that a sweep is the most likely outcome, in fact there is no case where that could happen in this year's playoffs. Pick a sweep only when the home team is 3.67 or more times better than the away team, or has at least a 96.2% chance of winning.

Pick the home team in five

This is a much more common choice, especially with a lopsided first round series. Pick this when the home team is between 1.27 and 3.67 times better, or has between a 64.5% and 96.2% chance of winning.

Pick the home team in six

Never! The sixth game is on the road, so any team likely to win it probably would've already won in 5.

Pick the home team in seven

While rarer, when two teams are especially close the most likely outcome is often the home team in 7 games. Pick it when the home team is between .97 and 1.27 times better, or has between 51.2% and 64.5% odds of winning.

What?!? 51.2%? What if it's 51.0%, do you pick the home team in eight or something??

Turns out, that when the home team is slightly favored the most likely outcome is that the away team wins in 6 games, even though it's smarter to pick the other team to win. How do you compensate for this? It depends on what value is assigned to each. Using TrueHoop's 5 points for series winner and 2 points for correct number of games the boundary is set at .944 times better, or when the home team has a 50.1% chance of winning.

Pick the away team in seven

Don't do it! If they can win in seven, they're more likely to win in game 6 when they're at home.

Pick the away team in six

If you're going to pick the away team to win, unless for some super weird reason the away team is waaaaaaay better. Accordingly, pick the away team in six when it's 1.04 to 2.38 times better, or has between a 48.8% and 85.8% chance of winning.

Pick the away team in five

There's only a brief range where this is the right choice, but it still exists. If the away team is 2.38 to 2.83 times better, implying between 85.8% and 90.3% chance of winning, then pick the away team in five.

Pick the away team in four

If the away team is 2.83 or more times better, meaning they'll win at least 90.3% of the time, then pick them to sweep. This has probably never happened, and probably never will.

Summary (for lazy readers):

If you think the home team will win pick them in 5 games, unless if you think it's close than go with 7 games. If you think the away team will win, pick them in 6 games pretty much no matter what. Importantly, while it's often tempting it's never smart to pick the home team in 6 or the away team in seven. There will of course be sweeps this year, and home teams that win in six, but from a purely mathematical perspective, that's not a smart pick going in.

89 Games

Congratulations to the UConn Huskies women's team on their 89th win, eclipsing the Wooden/Walton UCLA teams from the early 70s. I'll admit, I was rooting against them. Not, as Geno would suggest, because of some hatred for women's athletics but because as a Stanford fan the UConn Huskies are the Lakers/Yankees/Patriots of the sport.

Some people have made a big deal about comparing the records of UCLA and UConn and their 88 and 89 game win streaks respectively. UConn's streak is great, but it's true though Women's basketball has less competitive balance right now than Men's basketball. How does it compare to the UCLA teams, and even streaks in other sports? I can answer that.

I'm going back the the Random Chance in Sports stuff here, and examining how often the better team wins in women's basketball. The chances then of the best team winning 89 straight are .844^89. In men's basketball (by the same method) the best team wins 80.8% of the time, and in the early 70s they won 80.1% of the time (those numbers are close enough to be the same thanks to errors). To have the same unlikelyhood as UCLA's streak, the UConn women would have to win 115 straight games. Here's a look at how other sports compare (UConn column is what it would take to match the Huskies, UCLA is to match the Bruins):

	Uconn	UCLA	Longest
WCBB	89	115	89
MCBB	71	92	88
CFB	90	116	47
NBA	56	72	33
NFL	65	84	21
MLB	29	38	21
NHL	29	38	17

So the Huskies haven't yet matched what the UCLA men did, but their streak is far better than any other sports win streak. Maya Moore and company have played some incredible basketball, but Wooden's teams were amazing (7 straights NCAA championships!!). Now if you'll excuse me, I have to go make myself lunch in the kitchen.

A Comfortable Margin

I was watching a rerun of Winning Time on ESPN the other day, which was highlighted by Reggie Miller's 8 points in 9 seconds to beat the Knicks, all after Pacers owner Donnie Walsh had given up and retreated to his smoking room. The Knicks had been up by 6 with about 30 seconds left, was it reasonable for Walsh to leave? At what point in a game can you feel safe your team will win?

Just by observing basketball, a few years ago I had decided 6*(n)^.5 (where n is minutes left and n^.5 is the square root of n) was a good amount. If you were up by 6 or more with 1 minute left, 12 or more with 4 minutes left, 18 or more with 9 minutes left, etc. you'd be pretty safe.

Turns out the square root of n is pretty accurate. That's because the chance a team recovers a given amount of points depends on the standard deviation of the distribution of points. Standard deviation is the square root of variance, and variance can add. So the if the variance of 1 minute is V then for 2 minutes it's 2V, for 3 it's 3V, etc. So the variance is some constant k times n, and standard deviation is (kn)^.5, which is proportional to the safe margin.

Is the 6 constant accurate? Let's take a look at things on a per possession basis. The variance in one possession, based off of last year's numbers, is 1.22. In the final minute there are an average of 6 possessions total (more than the usual 4 or so), so the variance in the final minute is 7.3, and standard deviation is 2.7. Multiplying by 2.33 (so we're 99% sure, it's from z tables) we get 6.3, which is our final constant. 6.3!

Turns out 6*n^.5 was a pretty good guess, and accurate enough to use. It won't always work (it'll probably be a little less than 99% because one possession is not normally distributed), but it's a pretty good guide on when you should head back to your smoking room. Unless of course the other team has Reggie Miller.

New NFL and NBA rankings

Each week I have a chance to go over the college football rankings, but I haven't had a chance recently to look at the professionals. What's up in the NBA and NFL (besides me being embarrassed by the Detroit Lions)?

Here's four trends to look at going ahead:

1. The NFC West sucks.

Anybody who follows football already knew this, but having 24th ranked St. Louis as their best team is almost unbelievable. Ishan stands a chance at winning his bet that the 49ers make the playoffs (made when they were 0-5) but I think the winner is goint to be the Rams. That's right Masta' P, your St. Louis Rams will beat Seattle on January second for their 7th win of the season, clinching the division by tiebreaker over the Seahawks.

2. The Bears aren't just lucky.

Yeah, the Bears have won some close games this season. They've lost some too though, and updating the rankings through today they'd project to have 6.23 wins at this point, not too far from the 7 they actually have. They have a touch finishing schedule (average opponent rating of 1.67, right in between the Colts and Patriots) but with the parity in the NFL this year they are still predicted to have 2.73 wins. A 10-6 Chicago Bears team? That could make the playoffs.

3. The Heat will "get better".

Yeah, they're 7-4, but they're atop the power rankings and should be 9-2 (9.09 wins exactly). As the season goes on those mistakes will iron out, and I project them to finish at 67-15 at their current pace. They'll probably regress to the mean some, but the MiAmigos are still a very good team, probably the best in the NBA.

4. The Hornets are for real.

Chris Paul's return (not "CP3" a terrible excuse for a nickname) and the new focus on defense are a big deal. They boast the best record in the NBA at 9-1, and THEY'VE PLAYED THE TOUGHEST SCHEDULE. Once the easier teams roll in, they'll kick even more butt. At this rate, they'll finish the season 70-12! Don't expect that, but do expect them to stick around as a legitimate dark horse contender.

Random Chance in Sports

This post is a followup to Random Chance in the NFL which highlighted the lowered competition in the NFL this year. It showed that the best teams win 62% of the time this year, far less than previous years. How often the better team won in previous years and in other sports were still left unknown though. Until now. Here, using the same methodology focusing on variance, here are the data on the last three full seasons in the four major american sports:

	2008	2009	2010
MLB	0.59	0.6	0.59
NBA	0.77	0.77	0.75
NHL	0.56	0.61	0.61
NFL	0.8	0.8	0.78

On average the better team wins 60% of the time in MLB, 77% in the NBA, 60% in the NHL, and 79% in the NFL. But how about this year? How often does the best team win in the NFL this year? It's gotten better since the last look, so far this year in the NFL the better team wins 70% of the time. While this is still an unprecedented increase in parity, it is also slowly getting closer to the norm we'd expect, thanks largely to big losers like the Bills and Panthers. So while this NFL season may have some crazy outcomes, it's still less random than sports like Hockey.

This article originally ran with a bug in the program used to calculate variance. While the effect on previous season results was minimal, it messed with the current season results a lot. The NFL is not actually just random chance this year, although going with just home field advantage would still be pretty good.

How many for wins the Heat?

It's been covered, a lot. Everybody up through John Hollinger has projected the number of wins for the Miami Heat this season, and I have no secret methods, but I'm interested and there need to be some basketball articles for the NBA season. I looked at it two different ways. First, I added up last season's Win Shares of all the Heat players. Lebron had 18.5, Dwayne Wade had 13, and Chris Bosh had 9. Impressively, Udonis Haslem had 6 win shares last year (he's a solid defender), but that's where the help stops. For the season the Heat project to have a total of 67.1 win shares.

Some other historic teams:

08-09 Lakers: 60.8 win shares, 65 wins.

08-09 Cavaliers: 68.0 win shares, 66 wins.

07-08 Celtics: 70.2 win shares, 66 wins.

99-00 Lakers: 63.3 win shares, 67 wins.

96-97 Bulls: 71.3 win shares, 69 wins.

95-96 Bulls: 75.3 win shares, 72 wins.

It appears that win shares do a good job of predicting wi- Wait, THIS WON'T BE LEBRON'S BEST TEAM?!?! That's right, the 08-09 Cavaliers had 68 win shares, .9 more than this year's Heat project to have. The weak bench behind the Three Amigos will cost them this year.

The other way to analyze the Heat's upcoming performance is to look at their projected points. Last season they averaged 96.5 ppg, and gave up 94.2. Using Bill James's pythagorean method (winning percentage = ppg^14/(oppg^14 + ppg^14)), and updating projected ppg totals, it's possible to project win total. Adding Lebron and Bosh is a huge advantage here, bringing in 916 and 513 points each according to John Hollinger. The pythagorean method projects the Heat at 71.97 wins. One key difference between the two is Chris Bosh. He ranks the 8th best NBA player in Hollingers Value Added, but only the 19th best by win shares. Either way, high 60s is a good win projection for the Miami Heat this season. I like win shares better so I'll say the Miami Heat get 67 wins this season.

I'd also appreciate possible nicknames for this year's Heat. Andy, this is your time to shine.

Random Chance in the NFL

Normally over the course of an NFL season power rankings regress towards the mean and extreme ratings get closer to 1. This season however, the best and worse ratings are already closer to 1 than last season's final ratings, and we haven't had any undefeated teams for a couple of weeks. Has NFL parity really increased?

One way to judge parity might be to look at the percentage of time the better team wins. With no parity, the best team in a given game would win 100% of the time, being far better than it's opponent. But with perfect parity the better team would win only 50% of the time, and NFL game would be random chance. It's obvious that chance plays some role because otherwise the Rams would never win a game, but how much of a role it plays can be uncertain. To look at that, I'm going to look at what percentage of the time the best team in a game wins.

The key to my search was to look at variance in win numbers across the NFL. If all teams won 50% of the time, then winning percentages would be clustered around .500, giving a small variance. On the other hand, if the best team always one there would be undefeated and winless teams along with other extreme records, leading to a high variance.

Through 5 weeks in the NFL, variance per team was 1.24 wins per game. That matches up with the better team winning 62% of the time. To confirm, I looked back at the previous games to see how often the higher ranked team won. Turns out, the higher ranked team won 63% of the time, so it's pretty clear that in the NFL this year the better team wins somewhere around 62% of the time.

For comparison, I also looked at the last three years by variance. In 2009 the better team won 78% of the time, and in both 2008 and 2007 the better team won 80% of the time. If the Chiefs and Bears hadn't already made it clear, statistics show that this year parity and random chance have reigned over the NFL, so much so that picking winners is basically a crapshoot. At least that explains why my rankings systems can't pick NFL games. Also, it may give the Bears a chance at the playoffs. I kinda like this random chance thing.

Perfect Games

Poor Armando Galarraga. Today should've been a perfect game. However it'll still go down as one of the closest calls ever (Hey MLB, can we just fix the call and make it a perfect game?). Had Jim Joyce made the right call we would've had the third perfect game this season, and we're not even halfway done. What're the chances of that?! Funny I asked...

The key is on-base-percentage (we're not gonna count errors, dropped third strikes, etc.). There has to be 27 consecutive outs. Also useful is that there are 3,240, and about 345,000 all time. For the stats savvy I'm also assuming that perfect games follow poisson statistics, which can be a dubious assumption for low end odds.

The following stats are against MLB average, Yankees (best team), Houston (worst), Justin Morneau (best batter), Aramis Ramirez(worst, of course), and with each of the three "perfect" game pitchers.

Who	OBP		Expected		3+		All-time
MLB	0.33		0.065247621		4.40885E-05		6.966794621
Yankees	0.366		0.01468774		5.22314E-07		1.568278894
Houston	0.286		0.363409878		0.006106128		38.80297759
Morneau	0.487		4.82741E-05		1.86517E-14		0.005154448
A-Ram	0.227		3.100211665		0.598882668		331.0241437
Galarraga	0.218		4.237672039		0.794695585		452.476124
Halladay	0.258		1.02671629		0.08528064		109.627315
Braden	0.277		0.509658225		0.015130549		54.41859965

Because you get to have good pitchers against bad teams every once in a while, I'd suspect the odds of the number of perfect games are roughly equivalent to Houston (there are 18 perfect games in the modern era, plus a couple others that might've qualified in these stats), so we should have three perfect games in a year once every 200 seasons (actually as low as every 100 because poisson stats are kinda inaccurate here). That's more often than you think? Remember, there's now 30 teams in the league, many more than earlier, so more games. Also, this year's league average OBP is the lowest since 1992 (although roughly average overall. Can you say steroids?).

And in good news for Galarraga, let's suppose he pitches 30 starts a year for 15 year (a solid career). If he continues his opponent OBP of .218 (the lowest among league starters), he has a 44.5% chance of pitching another perfect game. Of course, with an OOBP like that, he'll also be the best pitcher of all time.

Good suggestion Greg.

Lebron A Bull?

Everybody from my Dad to Barack Obama has been campaigning for Lebron James to leave Cleveland and become a Chicago Bull. Today though, Jeff Van Gundy claimed that the Bulls still wouldn't be a real contender. So who's right? Fortunately John Hollinger has a statistic called Estimated Wins Added based off his PER stat. This is the number of wins (roughly) that a player would add over a replacement.

The Bulls should've won 36.1 games this year (They got lucky and won 41, but those sort of variations iron out over time). Luol Deng had an EWA of 7.4. Lebron's was 30.5. Putting Lebron in for Deng, and the 10-11 Bulls have 59 wins! Losing Salmons also will probably cost the Bulls an extra three wins from this year's team (he had a 5.9 EWA, against Hinrich's 1.0 over about 2/3 of the season).

What about other teams?

The Knicks (Lebron over Wilson Chandler) - 57 wins

The Nets (Lebron over crap)-46 wins

The Clippers (Lebron over more crap)-53 wins

The Heat (Lebron over Dorell Wright)-74 wins (umm, what? Okay, near extremes EWA isn't so good. Probably more like 65).

The Cavaliers (Lebron over Lebron) -59 wins.

So a simple Lebron substitution provides the following win totals:

Heat: 74 (65?)

Bulls: 59 (56?)

Cavs: 59

Knicks: 57

Clippers: 53 (more with Blake Griffin?)

Nets: 46

And purely for my amusement, the Bulls with Lebron and Chris Bosh?

+11 for Bosh. Bulls: 70 wins!

and Dwayne Wade (+20)?

Bulls: 90 wins!!!!!

Okay probably not. But it sounds like a record breaker. DWade for a midlevel exception?

The point is, JVG is right. For Lebron to really put a team over the edge he has to join a team that's already good (like the Heat) or he has to bring somebody like Bosh with him. He still helps though. I'm rooting for Lebron as a Bull.

Worst 5

In tune with the last post, here are the worst 5 teams of the last decade. Again, this is the old ratings system. This year's New Jersey team interestingly doesn't even make the list, as in the old rankings system they had a PR of .382. In the new rankings the 99-00 Clippers team has a PR of .206 (so again, the 1.5 rule roughly applies).

1	LA Clippers	1999-00	15-67	0.30162
2	Chicago	1999-00	17-65	0.329756
3	Atlanta	2004-05	13-69	0.333346
4	Cleveland	2002-03	17-65	0.333398
5	Chicago	2000-01	15-67	0.336922

Interesting to note here are the two Baby Bulls teams (they were bad for a while, a losing Dynasty of sorts), and the quick turn-arounds of Cleveland (Lebron!) and Atlanta (Joe Johnson?).

Top Teams Of The Decade

Something that I can't mispredict? The past! Here are the top 14 (14 includes the top team from each year) teams of the 2000's (99/00-08/09):

Note: This was on a slightly different (and less accurate) setup of the program I run, so the Power Rankings are a little off. Multiplying by about 1.5 is roughly accurate, as Boston's number one overall team had a Power Rankings of about 4.45. To compare, Orlando, this year's top team, had a PR here of 2.15 (The worst best team of the past 10 years!).

1	Boston	2007-08	66	16	3.01978476
2	Cleveland	2008-09	66	16	2.794060792
3	San Antonio	2006-07	58	24	2.702128779
4	San Antonio	2000-01	58	24	2.636983741
5	San Antonio	2003-04	57	25	2.627247807
6	San Antonio	2004-05	59	23	2.622031398
7	LA Lakers	1999-00	67	15	2.540923327
8	Dallas	2002-03	60	22	2.481384874
9	Sacramento	2001-02	61	21	2.424603144
10	Dallas	2006-07	67	15	2.322137833
11	LA Lakers	2001-02	58	24	2.290126862
12	Boston	2008-09	62	20	2.250968994
13	Sacramento	2002-03	59	23	2.246999335
14	San Antonio	2005-06	63	19	2.200287205

Interesting to note is the continuation of dynasties that shows up here, but doesn't necessarily show up in championship wins (hence, the appearances of the 2000 Kings and Dirk's Mavs). Is my algorithm a better way to determine champions than the actual playoffs? Do I just think because the Kobe Lakers don't appear hear? How will Tyler Dawson prove me wrong? These are important questions to ask. Hopefully, one other question has been answered here.

One Game Matters

The championship percentages I just put up don't include the first Cavs-Celtics game. What happens if it is included?
Before the series the Cavaliers had a 73% chance of winning the series. Now with their one win they have an 82% chance of winning. Had they lost, it would've been 53%. How does this affect championship odds? With the win they now win the NBA championship 27.7% of the time, as opposed to 18.0% if they had lost. So with just a Game One win (a game they were expected to win anyways) the Cavs have increased their championship odds by 53%. Which just goes to show: in the playoffs every game matters.

Unlikely Sweeps

As the playoffs progress into the second round, it may seem surprising that there's only been one sweep so far. It turns out though that even a heavy favorite is more likely to win in five playoff games then to sweep. For example, the most uneven first round series was the Cavs Bulls series. Based off of the power rankings (and a slight home court advantage) the Cavs were supposed to sweep 31.1% of the time (0.32% of the time for the Bulls). 35.5% of the time though they would win in 5 games. So despite all 7 True Hoop stat geeks predicting a sweep, a 5 games series was most likely. It was also what happened.

For the numbers intensive, what sort of power rankings lead to a sweep? Team A must be at least 3.67 times better than Team B for a sweep to be more likely than a four game series. That's a bigger difference than the difference between Orlando, the highest rated team, and Chicago, the lowest rated playoff team. That doesn't mean there won't be any sweeps this playoff, just that I won't be predicting any.