Mach Five

Nate Silver: Red Sox were a 278 million to 1 dog to be eliminated from the playoffs three times in 2011

2011-09-30T11:09:00.004-04:00

Nate Silver writes in FiveThirtyEight:

The following is not mathematically rigorous, since the events of yesterday evening were contingent upon one another in various ways. But just for fun, let’s put all of them together in sequence:

The Red Sox had just a 0.3 percent chance of failing to make the playoffs on Sept. 3.

The Rays had just a 0.3 percent chance of coming back after trailing 7-0 with two innings to play.

The Red Sox had only about a 2 percent chance of losing their game against Baltimore, when the Orioles were down to their last strike.

The Rays had about a 2 percent chance of winning in the bottom of the 9th, with Johnson also down to his last strike.

Multiply those four probabilities together, and you get a combined probability of about one chance in 278 million of all these events coming together in quite this way.

When confronted with numbers like these, you have to start to ask a few questions, statistical and existential.

Jesus. There's a difference between "mathematically unrigorous" and "a total sack of crap".

Penalties in soccer tournaments

2011-07-10T22:40:00.002-04:00

As far as I can tell, if a soccer match has to be decided by penalty kicks, many teams (men's and women's teams, club and national teams) simply ask for volunteers when filling out the lineup.

Call me naive, but I can't see how this is in any way optimal, either technically or psychologically. In this year's Womens' World Cup, the England side followed in the footsteps of the men's side by losing a shootout, and afterward the coach blasted the team in the press because nobody stepped up to volunteer for a penalty. In contrast, one of the coaches for the USA women's side watches the players take penalties in practice and picks the lineup from the available players accordingly.

Really, I can't see how asking for volunteers can possibly accomplish anything other than letting the players psych themselves out, and I'm shocked that teams at the highest levels of the sport do this.

Lance Berkman is my new favorite baseball player

2011-04-24T18:48:00.003-04:00

ESPN sideline reporter Wendy Nix: "You've hit more home runs against the Cincinnati Reds than any active player in major league baseball -- I don't know if you knew that. Why is that?"

Berkman: "I don't know. I think it's a statistical anomaly...I think it's just one of those things where everybody has a team that they seem to do better against, and then there are other teams, for whatever reason, you don't do well against, so I can't explain it. I've been asked that several times, but I have no legitimate explanation for it."

Random math fact of the moment

2011-04-07T19:23:00.003-04:00

The t-test is a statistical test used (among other reasons) to determine, given two samples drawn from normal distributions, whether the two distributions have equal mean. It was developed in 1908 by a chemist working for the Guinness brewery.

Dear Reggie Miller: Gus Johnson > you

2011-03-28T20:38:00.002-04:00

In Florida's overtime win over BYU in the round of 16 in the NCAA basketball tournament, Florida had the ball in the frontcourt with about 15 seconds remaining and the score tied. Gus Johnson asked if BYU should commit an intentional foul to try to get the ball back, and Reggie Miller reacted as if Gus asked him if the moon was made of green cheese. At the time Vernon Macklin, Florida's starting center (11.5 PPG in 24.5 minutes, 59.3% FG but 45.1% FT over 100 free throws or so) was on the floor, and Florida was in the double bonus (two free throws for every non-shooting foul).

I won't bore you with the math here, but suffice it to say that one can construct plausible percentages such that fouling Macklin quickly if given the chance is marginally the correct play. If Florida is in the one-and-one, it becomes a much better play. The better the offenses are, the worse the free throw shooter for Florida is, and the worse BYU's chances in overtime are (they looked pretty gassed down the stretch to this non-professional observer), the more the decision swings in favor of fouling.

What I find really interesting is that in Florida's next game, an overtime loss to Butler, nearly the same situation transpired. Florida had the ball with :30 left in a tie game. However, Florida coach Billy Donovan kept Macklin on the bench for the final possession. The only reason I can think of for this is that Donovan feared that Macklin might get fouled with significant time left on the clock, and it's quite possible that Donovan didn't think of this possibility until news of Gus Johnson's allegedly ridiculous question hit the interwebs.

I don't think it's correct to keep Macklin on the bench in the above scenario; I think it's better to have him on the line than to run a final possession without him. But it is very interesting that Donovan thought of this possibility and that he thought that Butler coach Brad Stevens would have thought of it also.

Props to Jay Bilas for not being results-oriented

2011-03-28T20:25:00.002-04:00

He stands by his claim that VCU didn't deserve to make the NCAA tournament.

Of course UAB, another team he panned, got blown out in the play-in round, but he correctly refuses to use that as post-hoc justification as well.

Knee-jerk reactions

2011-01-11T20:59:00.003-05:00

I don't know if our current gun control laws are optimal. I am fairly confident, though, that public shootings are sufficiently rare events that drafting new gun control laws in their wake is a bad idea.

The sabermetric revolution is coming to public schools

2010-08-18T18:02:00.008-04:00

LA Times and some guy at the Rand Corporation: Some elementary school teachers in the LAUSD are much more effective than others (as measured by performance on standardized tests), and that distribution doesn't correlate particularly well with the prestige of the school, seniority beyond ~3 years, additional degrees and training, or a bunch of other factors that people usually think are important.

Head of LAUSD teachers' union: LOL Rand Corp. are a bunch of idiots, the study is flawed because of (insert super-obvious criticism here), we're boycotting the LA Times.

P != NP

2010-08-09T21:49:00.001-04:00

So some guy at HP who appears to have a pretty good reputation has claimed to prove that P != NP. Not only am I relieved that I don't have to stop transmitting sensitive data on the Internet, I also get to post this screencap from an episode of Futurama which aired in 2000.

Transpositions and S_n

2010-07-04T11:52:00.040-04:00

According to official TV listings, there is an upcoming episode of Futurama called "The Prisoner of Benda" involving a device which allows people to switch minds. The title seems to be a riff off "The Prisoner of Zenda", a novel in which a fictional king is abducted and a distant relative who happens to look like the king is forced to impersonate him.

A few months ago, one of the staff writers for Futurama leaked the following photo of a script from the new season:

(Photo hosted by Futurama Madhouse.)

The page appears to be a lemma concerning the number of transpositions required to construct a permutation on the first n elements of the set {1, 2, ..., n, A, B} when one element of the transposition is one of the first n indices and the other element is one of the two auxiliary indices. Constructing permutations out of transpositions via algorithms such as selection sort and bubble sort is of course well-known; the only wrinkle here seems to be the auxiliary set.

It seems fairly straightforward to construct the denouement of the episode (or an approximation thereof) from this information: our heroes start fooling around with the mind-swapping device and somehow end up imprisoned. They discover that the device has a limited number of charges and need to undo the mind-swapping, but they're positioned such that one end of the device is in their cell and the other end of the device is in an adjacent cell inhabited by two other people/animals/potted plants/whatever. However, it seems to me that only one auxiliary element is needed to construct a k-cycle, and that it only requires (k+1) moves, so I could well be missing something here.

Here's the basic algorithm: assume we have a k-cycle which does not fix 1; otherwise this reduces to the (k-1) case. Write out the n-cycle in two-row form. For example, the permutation 2->4->1->3->2 on 4 elements would be written as

1	2	3	4	A	B
3	4	2	1	A	B

Swap 1 and A:

1	2	3	4	A	B
3	4	2	A	1	B

Swap A with whatever element belongs in the spot it's in now (in this example, 4):

1	2	3	4	A	B
3	A	2	4	1	B

Repeat the step above (in this case, swap A and 2), and continue until done. Because the original permutation was a n-cycle, each step will involve swapping A with a different element of the set {1, 2, ..., n}, and the final step will undo the original permutation, with A in its original place:

1	2	3	4	A	B
4	2	4	1	A	B

1	2	3	4	A	B
3	2	A	4	1	B

1	2	3	4	A	B
A	2	3	4	1	B

1	2	3	4	A	B
1	2	3	4	A	B

More formally, given a n-cycle permutation sigma on {1, 2, ..., n}, apply the transposition (A, 1), then (A, sigma^(-1) 1), then (A, sigma^(-2) 1), ..., (A, sigma^(-(n-1)) 1), then (A, 1) again.

Producer David X. Cohen claimed that writer Ken Keeler (who has a Ph.D. in control theory from Harvard) penned this theorem specifically for the episode. Calling it a "theorem" is a stretch (it's about the level of a homework problem for an introductory undergrad course in discrete math), but nonetheless it's amusing that Keeler managed to work it in.

EDIT, 8/19/10: The actual episode aired tonight, and the wrinkle is that once two bodies use the mind-swapper, that same pair can never use it again. In this case, two extra bodies are both necessary (we used the A, 1 swap twice above, so the above method won't work) and sufficient to undo any permutation on {1, 2, ..., n}.

Suppose the minds in bodies {1, 2, ..., n} are permuted by a cyclic shift:

Bodies	1	2	...	n	A	B
Minds	2	3	...	1	A	B

Pick some i strictly less than n, and apply the transpositions on the pairs of bodies (A,1), (A,2), ..., (A,i) in that order:

Bodies	1	2	3	...	i	i+1	i+2	i+3	...	n	A	B
Minds	A	3	4	...	i+1	i+2	i+3	i+4	...	1	2	B

Bodies	1	2	3	...	i	i+1	i+2	i+3	...	n	A	B
Minds	A	2	4	...	i+1	i+2	i+3	i+4	...	1	3	B

Bodies	1	2	3	...	i	i+1	i+2	i+3	...	n	A	B
Minds	A	2	3	...	i+1	i+2	i+3	i+4	...	1	4	B

...

Bodies

...

i+1

i+2

i+3

...

Minds

...

i+2

i+3

i+4

...

i+1

Now apply (B, i+1), (B, i+2), ..., (B, n) in that order:

Bodies

...

i+1

i+2

i+3

...

Minds

...

i+3

i+4

...

i+1

i+2

Bodies

...

i+1

i+2

i+3

...

Minds

...

i+2

i+4

...

i+1

i+3

Bodies

...

i+1

i+2

i+3

...

Minds

...

i+2

i+3

...

i+1

i+4

...

Bodies

...

i+1

i+2

i+3

...

Minds

...

i+2

i+3

...

i+1

Now apply (A, i+1) and (B, 1):

Bodies

...

i+1

i+2

i+3

...

Minds

...

i+1

i+2

i+3

...

Every swap takes place between one element of {A,B} and one element of {1,2,...,n}, so the process by which {1,2,...,n} got swapped is irrelevant, and none of the swaps in our procedure are duplicated.

If the set {1,2,...,n} forms the only cycle that needs to be unpermuted, swap A and B. Otherwise, apply the same procedure to all disjoint cycles in succession. If there's an odd number of disjoint cycles, then A and B will need to be swapped at the end.

EDIT, 8/20/10: Welcome both of you io9 readers. Out of concern that somebody might actually read this, I've expanded the explanation of the algorithm slightly.

Note to David Ortiz

2010-04-20T23:46:00.002-04:00

Nobody's too good to have to fight for playing time. Mike Lowell and Jason Varitek realized this, perhaps because they didn't receive a two-week grace period at the beginning of this season.

"Honors is epaulets. Honors is uniforms."

2010-03-20T17:48:00.001-04:00

Life imitating Futurama

2010-03-09T21:31:00.003-05:00

A couple days ago, I got a letter from the government informing me that in about a week, they would be sending me a 2010 Census form. Obviously someone from the Department of Commerce is a Futurama fan.

"Attention Hermes Conrad: you are about to receive a letter from the Central Bureaucracy."

More LOL donkaments

2010-03-03T21:46:00.002-05:00

These are consecutive hands from a $33 heads-up match (the dollar signs in the graphics represent chips, not actual cash, for the uninitiated).

As an additional insult, Villain realized his shame and declined a rematch.

Fan noise during sporting events

2010-02-27T18:55:00.004-05:00

Apparently in curling, as in tennis, golf and some other sports, it is considered proper etiquette for fans to be silent while the players are in action. Various breaches of etiquette during the Vancouver Olympics have created somewhat of a kerfluffle.

Why exactly are the peanut galleries in these sports supposed to be quiet? It seems to me that noise per se shouldn't really be much of a problem for players in any of these sports. Sudden differentials in noise volume, however, would be a significant problem, and demanding that fans be silent exacerbates the problem -- it's a lot easier for a few obnoxious/ignorant fans to create a distraction when the rest of the crowd is silent than it is for a few thousand fans to suddenly start/stop cheering in unison.

Career passer ratings

2010-02-08T09:31:00.007-05:00

Fun game: Try to identify these NFL quarterbacks (all of whom have played in the 2000s) from their career regular season home/road passer rating splits (all statistics taken from pro-football-reference.com:

Quarterback A: 98.9/91.6
Quarterback B: 96.8/90.9
Quarterback C: 97.7/85.9
Quarterback D: 94.7/89.0
Quarterback E: 99.1/92.1
Quarterback F: 96.1/95.6
Quarterback G: 89.0/84.4
Quarterback H: 91.9/91.6
Quarterback I: 94.2/92.5

Answers in white below:

Quarterback A is Peyton Manning.
Quarterback B is Kurt Warner.
Quarterback C is Matt Schaub.
Quarterback D is Drew Brees.
Quarterback E is Tony Romo.
Quarterback F is Philip Rivers.
Quarterback G is Brett Favre.
Quarterback H is Ben Roethlisberger.
Quarterback I is Tom Brady.

For the alleged quant-fans who think that Quarterback A is the greatest quarterback of all time -- can you think of a reason why Quarterbacks A-E have such pronounced splits?

Ladbroke's welches on a parlay bet

2010-01-13T00:02:00.002-05:00

Apparently they took a parlay bet on snowfall on Christmas day in various cities around the UK and lost. Hooray for correlation!

Big Brother is watching

2010-01-12T19:15:00.003-05:00

And by "Big Brother", I mean a bunch of punkass kids fresh out of Harvard and Stanford.

Fun poker facts

2009-12-14T18:58:00.004-05:00

I just played a heads-up match against a fella who had a -33% ROI (return on investment) over 2,000 matches. I'd say that's probably statistically significant.

To put that in perspective, 32o stands roughly a 32.3% chance of winning against a random hand, according to PokerStove. Let's just call it 33%. So if he gets dealt 32 offsuit every time he plays heads-up, and always turns it faceup and pushes all-in preflop, he'll win 1 out of every 3 times and end up losing a total of 1 unit for every 3 invested, which is a ROI of -33%. Hilarious.

More bluffing with the best hand

2009-09-16T22:15:00.003-04:00

Straight draws are nice hands with which to bluff/semibluff because of the combination of fold equity and suckout equity. That being said, I was pretty sure I was toast when Villain (25/14 pf) didn't fold the turn.

LOL donkaments

2009-09-16T00:15:00.003-04:00

I just booked a wager with DA, one of my new coworkers. I'm laying 5:1 odds that Phil Ivey won't win the 2009 WSOP Main Event. The best part of the negotiations was when I proposed a small wager size, and DA immediately countered with a significantly larger wager. I feigned nervousness and swallowed hard before accepting the offer.

Anywho, although I am certainly tempting fate by opposing Phil Ivey and posting about it, the image below makes me feel pretty good about my chances.

"Everyone kills people, murders people, steals from you, steals from me"

2009-09-06T09:10:00.003-04:00

So says Ohio State quarterback Terrelle Pryor. (Scroll ahead to 2:45 if you're impatient.) I guess if that's the case, then running a dogfighting ring is like TPing someone's house in comparison.

Anyway, there's a good reason college teams often don't let their players address the media solo. That being said, I'd like to think that if I were one of the reporters there I'd have given Pryor a chance to retract or amend his statement, but oh well.

These people aren't your friends; they're paid to kiss your feet

2009-09-06T08:25:00.002-04:00

From Marginal Revolution, we learn that some marketing company in Australia is selling "friends" to Facebook users.

I have to say that $177 for 1,000 friends is rather expensive, though. For that price, I'd expect them to attend my wedding as well.

This guy is either on super gorilla tilt, or he read my soul. Possibly both

2009-09-04T22:16:00.003-04:00

Villain here is 18/13 preflop, and I had 3bet him a couple times and forced him to fold preflop.

Bluffing in 3-bet pots is fun. Especially with the best hand

2009-08-24T22:01:00.001-04:00