Dogs

The following conversation actually occurred during my class today.

Student 1:  A calculator is a man's best friend.
Student 2: I thought that a dog is a man's best friend.
Me: Ah, but since you can program a dog into your calculator, the calculator is really your best friend.

I then proceeded to enter aforementioned program into my calculator, which reads as follows (in calculator-language):

4->A
While(A>0)
Disp "Food?"
Pause
Disp "Play?"
Pause
End

For those who don't speak programming, this creates an infinite loop where the calculator dog asks you alternately for food and to play.  I thought it was pretty accurate.  I also am very happy that Kaylee is much more sophisticated than a dog:

She also sleeps where my feet go.  Bravo Kaylee.

 

 

 

Math and Mathemagic

I had a fun lesson today in Statistics.  We just started talking about experimental design and actually how the statistics works (fuck off linear models, we're moving to important things now) and I really wanted to leave a good impression for when we start talking about confidence intervals and p-values.  So I came up with an activity to do.

The big question of the day that I wanted to answer was, "How do we know that an event is due to factors OTHER than random chance?"  The best example I could think of was a question I had on a statistics test.  "How many times would you have to a six on a six-sided die out of 120 rolls in order to be 95% confident that the die is loaded?"  Now, since I'm a good gamer, I don't have any loaded dice.  But I can force a card.

Pick a card, any card....

 

So I forced the card and then later in the lesson I revealed the card.  How can we be certain that I wasn't just plain lucky?  It spawned a fun discussion and realization that, no matter what the situation and what the sample, there's a small random chance that the unlikely happens.  And thus, we never "prove" or "disprove" anything with statistics.  We can only be 99% confident.

Statistics is fun.

My mind is melting....

I was getting my Daily Show on.  Because it's the weekend and I have time.  Now, sometimes Hulu says, "would you like to watch the long-form commercial of something or another instead of the normal commercial breaks?"  Normally I say no when I'm watching a 30 minute show on the off chance that I get a 2.5 minute long commercial when normally I only get 1.5 minutes of commercials (I'm a mathematician; I can't help it).

Hulu very politely asks me, "Would you like to watch the Kohler advanced toilet long-form commercial?"

Like the questions "Do you want to smell this?  It smells awful," and "Can your farts save your buddy from drowning," (If you haven't watched at least one episode of MANswers, you should give it a shot just to see how dumb the male species is) this question only has one answer:  Fuck yes.

So I'm watching in awe as a camera pans around a bathroom which is nearly the ballsiest I've ever seen (the first being in my grandfather's house).  In the middle of the room sat the toilet, and on three sides were full room-height windows.  "Lid and seat open and close automatically," drifted across the screen.  "Fuck me," thought I, "this device can prevent 25% of all divorces in the United States!"  Next came, "heated seat" and "warms your feet."

But wait, there's more.  This toilet comes with a wireless touchscreen remote control.  If you haven't passed out from shock, perhaps you didn't hear me.  A WIRELESS TOUCHSCREEN REMOTE CONTROL.  FOR YOUR TOILET.

All this can be yours for the low low price of $4,000.

Kepler Candy

German Mathematician/Astronomer Johannes Kepler

A student of mine presented a problem to me today.  His physics book was using Newton's 2nd law to derive Kepler's third law of planetary motion.  The authors substituted in 2πr for the distance a body would travel in one orbit.

"But wait!" said student, "my buddy Johannes says that all orbits are ellipses, not circles!  What are the authors trying to pull?"  Student concluded correctly that the authors were using circles to model the elliptical orbits, and so the question he asked me is, "How accurate is this model?  Why do we use it?"

To answer the second question, it's because the equation for the circumference of a circle is a lot easier than the equation for the circumference of an ellipse.  Don't believe me?  C_c=2πr.  On the other hand, C_e is:

Equation of an Ellipse...

But wait!  There's more!  What's this E(stuff)?  Well Doubting-Sceptic-Who-Still-Thinks-This-Is-Simple, this is called the "complete elliptical integral of the second kind."  Sounds complicated, huh?  It's because it is.  Take a look at it:

There's a reason why you didn't learn this in school.

So, now that we know what C_e and C_c are, we need to figure out how closely one approximates the other.  We also need to look at how big our approximating circle is.  First off, let's call the distance from the circumcenter of an ellipse to the edge the fake-radius (or "radius" for short).  We want our average "radius" to be somewhat close to the radius of our model circle.  Let's start by figuring out the average "radius" of an ellipse.

So....we're going to be adding a bunch of "radii" together.  We have an uncountable number of distances.  Oh noes!  What should we do?!

I'd call integral man, myself...

 Integral man to the rescue!  Hooray!  Let f(Θ) be the function which, given an angle Θ, gives the distance from the circumcenter to the edge of the ellipse at the point of intersection.  Then the average "radius" would be:

Let's take a quick look at what f looks like.  It's going to start at (0,a), head downwards to (π/2,b), then go back up to a, go down to b, and back up to a.  Oh, wait, that sounds familiar!

Bam!!!  cos(x)'d

 Now, keep in mind that cosine models a circle.  The exact shape doesn't matter...what does matter is the symmetry on a horizontal and vertical axis.  Now, the integral of cos(x) is 0 on this interval.  So the average value is also 0.  Our graph, displaying the same properties of symmetry, is exactly the same, but translated upwards by (a+b)/2.  So our average "radius" is (a+b)/2.

So now we're comparing C_c=2πr=π(a+b) to C_e, which was still a mess.  But things get a little bit better.  Check out this approximation for an ellipse:

Look at that π(a+b) that's just begging to be canceled out!

 So if we look at C_e/C_c, we simplify the mess to be just the mess equation in the parentheses.  Since we're looking at how accurate the circle is modeling our ellipse, let's graph this and see how different values for a and b affect our model.

Thanks Conrad!  You're a pal!

 Since we're looking at differences, we're only looking at positive a and b (top right corner).  Since we're looking at a ratio, we want this to be close to 1.  As we can see, it looks that no matter the size of the ellipse, we have a relatively close approximation, using the circle, to the circumference.

I hope this satisfied the math appetite you built up during my 2-month hiatus.  Special thanks for the folks at WolframAlpha/Mathematica for giving me free 3d graphing technology.  You guys are the best!  <3

An early In Which Kaylee Decides

I'm in Pittsburgh now, preparing to go hide in Amishland, PA for a week and gathering all of my belongings in preparation for a move to Oklahoma City.  Because I'm a heartless bastard, I had to leave Kaylee behind with a boarding service.  She didn't want to be left behind.  She wanted to come with us:

In Which Kaylee Decides to Come to Pittsburgh (click to embiggen)

I miss my kitty.

This week in In Which Kaylee Decides

I doubt this will become a weekly thing. But I do hope so. We can at least pretend that it is a weekly thing. Without further ado, here's this week's In Which Kaylee Decides:


In which Kaylee decides to sit in a trash can:

Rolling Dice

Everyone loves to roll dice.  It's in our very nature (well, at least my very nature).  However, sometimes rolling dice is not the best tactical decision.

I was over at Miniwargaming's website watching the Tyranids and Salamanders slaughter each other in an epic 3000 point Apocalypse mission (Eldrad was there to make sure that the terminators were all rolling ones for their armor saves).  Coming down to turn 7, each player held three objectives.  The entire board, once filled with swarms of gargoyles and termagaunts, had been whittled down to one unit per objective.  However, in one corner of the board, the Tryanids were launching an assault.  It looked like this:

That rhino is full of marines bee tee dubs.

After careful consideration, the Salamander player decided to disembark his marines, move them forward to block the movement, and rapid fire into the gargoyles.  He was reminded that both the warrior and the gargoyles were winged, so any left over could fly over his marines in order to get to the objective.  It should also be noted that since this was Apocalypse, all vehicles are scoring, so he still was holding the objective.

Fracking toasters gargoyles.   

This allowed the last Tyranid Warrior to fly next to the rhino, assault it, and contest the objective within only a couple of millimeters to spare.  On a side note, let this be a lesson to all you Space Marine players; rhinos are just under three inches wide.  Don't let this happen to you!

  Objective contested!  All ur biomass r belong to nids!

This was a bad tactical decision on the part of the Space Marine player.  Because the Warrior was hiding behind cover, there was no way that he could have removed both threats in the same turn, especially while he had to stay close to the objective.  As much as I despise Space Marines, they do do one thing exceptionally well; they're very hard to kill.  And you need to take advantage of that.  He should have realized early on that there was no way that he could remove the threat.  However, there are other things that you can do in order to solidify your position without rolling dice.

Here are some rules to keep in mind:

1. Enemy units, unless assaulting, must remain 1" away from your units.
2. Jump infantry, while they can move over units during the movement phase, follow all movement rules for infantry during assault.  They cannot move through enemy units and can only move through gaps wider than their bases.
3. To initiate an assault, the closest two models are moved together first.

Keeping that in mind, you have several objectives you need to accomplish during your main phase.  First, you have to make sure that the pesky flying type units cannot land behind you to contest the objective.  In essence, this means that you have to force units out of the 3" capture zone around the objective by spreading out your marines.  Second, you have to make sure that no models can move through your lines in order to come into base to base contact with models in the back.  This makes it so they can't contest just with an assault move.  Third, you have to make sure that your models are staying close enough to the objective to hold it at the end of the turn.  This is my solution.

If you squint, it looks like a turtle!

With this setup, if the marines are assaulted, then their pile-in move leaves the second row of marines to be just holding the objective.  The rhino is in a much better position because it's forcing any other units from that side at least 5 inches from the objective.  I've also placed some marines behind the rhino in order to try to block a consolidation move if the rhino is destroyed.

As I've mentioned, this takes out a lot of the probability and randomness of the game.  Which is great if you want to win.  The first person's strategy depended on being able to kill all of the gargoyles, which had a decent probability of succeeding.  Nevertheless, if there is a way to win the game without rolling any dice, you should take it.

As an added bonus, 7 of the marines were in rapid fire range of the gargoyles in my deployment.  Which is still a decent number of dead gargoyles.

Patrick Rothfuss

I hope you read his blog. In fact, I hope you read his books.  If not, you should. Anyways, in his latest post he says:

They e-mail me with their theories and their hopes. They want X to hook up with Y. They want Z to get his comeuppance. They want such and such story tied up in a certain way….

As a mathematician, it irks me to see variables unsolved for. So I'll solve for Z, just for fun. I'll leave X and Y for now because it's the summer. And I'm lazy.  See: unsolved Price is Right problem.


Theorem: Z=Ambrose. ■

Proof: Duh. ■

Nothing Good Happens After 2 a.m.

It is further into the wee hours of the morn than I might care to say.  And in general, though I occansionally write in a sleep deprived stupor, I am hesitant to share what I've written, or even to read it myself, after a good night's sleep.  I will say that this one is not my fault.  Damn you Patrick Rothfuss.

Since I've been home, I realize that I've been passively longing for the...perks....of living here.  I sat down nearly twenty two hours ago to finish reading Rothfuss' "new" tome, The Wise Man's Fear.  It is a hefty behemoth of a novel nearly a thousand pages long and written in pure gold.  Regardless, this is not the point of my thoughts...merely the vessel.  This afternoon (well, yesterday afternoon if you are to believe the clocks) I witnessed several truly remarkable events which I had always taken for granted while living at home.  It's common knowledge to our family that a red-tailed hawk prefers a certain electric wire as a perch; this happens to be a scarce twenty feet from my window.  While the hawk was absent from the perch, her voice was not.  It was a delight to sit in my room, listening to the screech of the hawk (and while screech is commonly used to describe the noise a hawk makes, I hesitate to use it as it denies the hawk a true description of her voice....I shall have to think on this more when I have more wits about me).

Around midnight, the coyotes started barking and howling.

The most uncommon event of all happened around 3:20.  In all my family's years of living at the house (which isn't too many, I suppose...but it is worth noting that I am the unique member to have witnessed this piece), only one similar sighting has happened.  I was taking out the garbage one night, probably about nine years ago.  The sun had set, leaving only a dim glow behind the mountains and a colorless world on the front range.  I looked up from my menial task and there, sitting atop an evergreen as a star on a Christmas tree, formed the unmistakable silhouette of a great horned owl.  This morning gave me a second "sighting," when outside my window I heard the soft, piercing hoots of an owl.  Whether or not this was the same owl (or even the same species) I haven't the foggiest idea.  But I'm a romantic at heart, and it's good to know that my owl has stuck around.

In addition to delving myself deep in the Rothfuss' text and keeping an ear pressed to the window, I also kept an eye on my cat through all of this.  She gets frighted by engines.  By garage doors.  By my dad tromping up the stairs.  Yet even though I know that at least some of these sounds must be unfamiliar to her, she barely heeded any of the signs I heard.  I knew that in a fight, she could not come out unscathed versus a hawk or an owl.  I shudder to think of what would happen if she stumbled into the pack of coyotes.  Yet she ignored the hawk, played through the coyotes, and slept through the owl.  In fact, the only animals that held her attention all day were several moths at the window screen (which she tried to swat, even through the barrier) and a rather noisy bee, also at the window screen.  However, I noticed that she kept her distance.  Rather than sitting on the windowsill to watch the bee as she did with the moths, she sat on the chair next to the window to watch.  And she didn't swat.  Given that she was adopted about 6 months into her life and spent the next 11 months as an indoor cat, I would think that she would have no experience with bees.  I wonder if this is instinctual behavior or learned behavior....

Goddammit.  I've turned into a biologist.  And animal behavior seems to be my newfound passion.  Ted Mosby was right....Nothing good happens after 2 a.m.

Packing

I know it's been a long time since my last post.  Forgive me.  I was finishing my Master's "thesis."  Then trying to find a job.  And recovering.  And moving back home (with my parents....yuck!).

Anyways, the Price is Right problem is being put on the back burner for now, because it's an involved problem and I don't feel emotionally ready to tackle the whole thing just yet.  So instead, I present a problem on packing.

Traditionally, packing spheres into a three dimensional space has proved to be 1) very inefficient and 2) difficult to solve.  I was having this exact problem today when I tried to pack some 2-dimensional spheres (ok ok, I'll call them rings...) onto a rectangular tray.  You see, this is very important, because I need to maximize the number of rings in order to obtain the maximum number of Utiles (ah, you didn't know I was going to throw economics jargon at you did you?) from this particular exercise.

You see, anybody who knows anything about something must know that onion rings are delicious (the proof is left to the reader).  And if you're going to go through the trouble of making onion rings at home, you know that you have to maximize the number of breaded goodness that comes out of your oven.  So here I was, making a delicious batch of onion rings for myself, and here's my original solution to my "ring packing problem."

NOT ENOUGH!!!

  At this point, however, I still had some rings of onion to place on the baking sheet.  And then the most obvious solution to the "sphere" packing problem dawned on me......

Nest Them!!!!

 You're welcome, math.




p.s.  The recipe I use makes delicious onion rings without having to go through the both of deep frying at home is Chef John's recipe (over at Food Wishes) with the following changes:

1) No Cayenne Pepper.  It doesn't go well with the cheese.
2) Reduce the cooking time by about 2 minutes.
3) Once the rings are "done" cooking, sprinkle Parmesan cheese over top (if you notice, I put down some foil on the pan...this makes it a lot easier since you don't have to worry about getting the melted cheese off later).
4) Place the rings under the broiler for about 2 minutes to allow the cheese to melt and the breading to completely brown.

The Price is Right (Part 1)

I've been watching a lot of the Price is Right over spring break.  Some people really take this game seriously and study in order to really perform well on the show.

I'm looking at yoooou Barney!

Now, if you do seriously study for the Price is Right, there are two places where your game can loose clout.  First is getting George to tell you to come on down.  Second is the Big Wheel, because no matter how hard you study, it's a completely random process and there's no way to "guarantee" a win.  However, you can maximize your chances of winning.

Assume you don't spin first.  Then there is a bid that your spin must beat, and there is little choice on the contestant's part.  Thus, we assume that you spin first.  Now we need to figure out what number you should try to get above in order to maximize your chances of winning.

Since the wheel has values from 5 to 100, there are 20 spots for the wheel to stop.  The goal is to get closest to 100 from one or two spins without going over.  If your final total is a 50, then either person spinning after you can beat that score to knock you out (to simplify things, let's ignore ties for now).  For each person, they can either spin above a 50 (10/20) OR they can spin below a 50 and then spin a number which will not put them over.  I will consider each individual spin in order to correctly get the probabilities: if your first spin is a 5, then you have to get at least a 50, but not 100 (or 1/20*10/20).  If your first spin is a 10, you have to get between a 45 and a 90 (or 1/20*10/20).  If your first spin is a 45, then you have to spin between a 10 and a 55 (or 1/20*10/20).

This means that if you stay at a 50, then your chances of the first person beating you is (10/20+9*1/20*10/20=29/40), or almost 3/4.  The chances of either person beating you (roughly speaking) is 3/4+1/4*3/4, or 15/16.  We need a better number if want to get into the Showcase Showdown.

Continued in Paaaaaaart twwwwwwwo....

The Social Sciences

I was playing on Wikipedia the other day (basically looking up dropout ages for different states) and I stumbled across a certain philosophy (coined by one John Holt) which struck my interest: the unschooling movement. Basically, this guy wanted to completely do away with compulsory schooling. He said that students should be able to learn what they wanted to learn, when they wanted to learn it. Moreover, he says that it should be the responsibility of the parents to make sure that their kids are being successful, inquisitive learners.

Now here's my problem with the social sciences, and philosophy in particular. It seems that whenever a theory is presented, it's always presented in a perfect world model, and they never address why it might actually be a bad idea.

I bet you can guess which side I am on.

Here's the first issue that I have; he expects that parents give a shit about their spawn. Talk to any teacher about things they wish they could change about their school system, and I guarantee you that "I wish parents gave a shit about their spawn" would be in the top five. It's not a realistic assumption unless the government can somehow regulate it (which would require major changes in our constitution in order for it to be even legal, much less enforce it, etc.).

The second thing that is glaringly wrong with his argument is that we have already tried non-compulsory education. And it sucks. It created a rift in society where those who were educated had a huge advantage over those who weren't (granted, education was also a mark of wealth, and so part of it was that the rich were educated, not that the educated were rich). I mean, just think for a second about how much money casinos make because of their mathematicians. The mathematics of calculating probabilities is certainly within the grasp of most high schoolers, and yet these people still go to casinos to play losing strategies.



Lastly, schooling provides some students the opportunity to rise up out of their current social situation, and they grow up to be extremely successful.  My grandfather grew up in rural Nebraska and was recruited by West Point out of high school and afterwards was very successful (in the American Dream sort of way).  Had he been a product of the unschooling movement, I have little doubt that he would have continued working at his parents' gas station in Nebraska until he was old enough to take over.

 Now, that's not to say that our current system is better (in fact, it is far from perfect).  But in the end, this is, in my eyes, a distinct step backwards.

Final Four

 ...and I was worried when my bracket didn't have any number 1 seeds going to the final four.

UCONN at the moment has the best chance of the teams left, and I'm not just saying that because of how the seeds worked out.  The dice chose UCONN.  Which means they have always had the best chance of winning the tourney.

The dice also choose to form impossible shapes

In all seriousness, I've been thinking that maybe this whole method is not actually as good as I want it to be.  The NCAA is really bad at choosing seeds.  I mean, sure you can get some information out of the seeds, but really the seeding makes some teams just have an easier path towards the end of the tournament.  Given the assumption that a three seed (like UCONN) is just as good as a one seed (i.e. each has a 50/50 chance in winning...which is an assumption I made), the only thing that makes a difference is the path that the teams take to get to the championship.  UCONN is going to be facing better teams earlier than, say, Ohio State did.

 Furthermore, the sheer number of possible outcomes makes it so that it is nigh impossible to get a bracket which is good (on the other hand, the sheer number of people making brackets makes it so that somebody is going to have a good bracket *wink*).  Think about it this way...instead of playing basketball, each team decided to flip a coin to determine the winner.  The model should be pretty clear...each team is going to have a 50/50 chance in winning.  Given that there are five rounds of play, the probability that you pick the correct winning team is (1/2)^6, or 1/64.  The problem here is that all of these probabilities are being multiplied together.

Warhammer fans should be familiar with this decay; a space marine squad rapid firing into another space marine squad has 2/3 chances to hit, 1/2 chances to wound, and then 1/3 chance for the opponent's armor to fail.  These aren't awful odds for this situation, but when you multiply it all out, you're only expecting 1/9 shot to wound.  That means that on average, from 20 shots, you're killing 2 marines a turn.  Not exactly stellar odds, but because three things need to happen for you to kill a marine, you're multiplying all these numbers together which are less than one, and they get small quickly*.  So no matter what the odds are for a 1 seed to win the championship, the probability is very slim that it will happen because of the layers of events which need to happen in succession.  Oh, and unlike Warhammer, you only get one shot at this one.

U Newbie!!  I kill you!!...Or...maybe not....

Now the interesting question becomes...can you do better than the average bloke will with a bracket rolled with dice?


*Side note: this is why things like Fortune, Feel No Pain, Ghosthelms vs Perils attacks, etc. are so freaking good....they add more dice which need to be rolled to kill your guys.  This is also why power weapons and other armor-ignoring weapons are so fantastically good...they subtract the number of dice which need to be rolled to kill other guys.

Taco Bell

They are now serving shrimp.  I have spent the last six hours carefully analyzing the probability that this will not give the eater some sort of disease.  Drum roll please:

<.01%.  I base this on the fact that I am throwing up in my mouth right now.  The picture is obviously infecting my brain.

That is all.

Perudo

Basically the game liar's dice from Pirates of the Caribbean.  Except you aren't giving this man your soul when you lose:

Do you fear dice?

My friends hate it when I play this game.  Because I think too much, and it's a drinking game.  But the whole concept of the game is fascinating.  There's a total of 5 dice per player (so with 5 players, you start with 25), only 5 of which you know.  You have to guess with strictly increasing bets how many dice there are with a certain number or a one.  Therefore a bet of three threes means that you think that under all the cups, there are at least either three threes, three ones, or a pleasant combination of the two.  An average roll at the beginning of the game will net you about 8 each of two's and one's, three's and one's, four's and one's, etc.  But the whole idea is to use your dice in order to give yourself an advantage: there's a lot more chance for your five dice to roll abnormally than the other 20 to roll abnormally.

The betting system is thus: each subsequent bet must either increase the number of dice with a certain number OR increase the number on the dice but keep the same number of dice.  So if I say there are four threes, then a legal bet would be either five threes or four fours.  However, 2 threes is right out.

And he shall snuff it....

 After each bet, the next player can call bullshit, bologna, liar, scurvy dog, or some other appropriate (or inappropriate) phrase.  If he does not, then he must make a bet.  If bullshit is called, all dice are revealed.  The loser removes one of his/her die, and the game continues.

Ones are special.  The initial bet must halve the number of dice bet previously (rounding up).  After ones are started, there are two separate betting tracks, one for ones and one for everything else.  They do not affect each other.

Now the trick of the game, after playing several times, is to force an opponent to make a bet which is just completely awful and ludicrous.  This means that if I am holding five of one side in my cup, I'm betting high just so the betting doesn't make it all the way back to me.  It should go without saying that the more dice you have, the better advantage you have.

 The second trick is to recognize when someone has you beat and you need to analyze your options.  Picture this: there are three dice left.  Two fours are bet by your opponent.  You're looking at a three in your cup.  There are a couple bets you might consider.

1.  Bullshit.  Think about your opponent.  The probability that they have two fours (or two ones, or some combination) under their cup is 1/9.  This is not as simple as it seems.  The probability of rolling "doubles" is 17/36, or just under 1/2.  Does your opponent seem like the lying type?  If so you're in luck.  Does he seem like the play it safe type?  Well then maybe you want to choose another option.

2. Three fours.  You're only putting your opponent into a pickle if he actually *has* two fours.  I hope you're a good liar if you choose this one.

3. Three threes.  You already know there's one.  So of the two unknowns, both have to be threes.  1/9 probability of winning.

4.  One one.  Either there's a one on his first die or a one on the second.  1/6+1/6.  1/3.

Here, bullshit might be your best option, depending on who you're playing.  A lot of times when I'm starting out rounds with 10 or fewer dice I just spout out "One [insert random number here]."  Second best option is clearly one one.

Now the challenge here is to be able to do this in your head.  While drinking.  Do it.

Make your friends some of these so they are placated while you think.

March Madness

I have a strange fascination with statistics and probability.  As such, rather than completely filling out a bracket randomly, or just going by seed, I decided that the best option was to roll dice.  I figure that a good goal will be to get 50% of the games correct.  This might be a project for next year where I do several hundred brackets to find an average winning percentage and a standard deviation.

In short, I've created a chart.  If the seed of the opponents are within 2 (a 3 seed vs a 5 seed, etc), then each has a half of a chance of winning.  Between 3 and 5 seeds apart, the lower seed will win on a 7+ roll of two dice (best two of three).  6-8 is 6+ (best two of three), 9-10 is 5+ (BTOT), 11-12 is 4+ (BTOT), 13-14 is 3+ (BTOT).  There is zero probability for a 16 seed to beat a 1 seed.

So I rolled up a bracket.  UCONN should win.  So sayeth the dice gods.  If I were less lazy I'd find the probabilities for you and make a nice graph to post to find probabilities of seeds winning games.

In other news, I will never get tired of CSI Miami's one liners.

OMG!

OMG I have a Blog!