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The Nation Has Problems 6

GreekHouse November 26, 2007, at 8:00 pm 40 LTEs - (Sorry, but the time to add LTEs has expired)

TNHP went on vacation last week for Thanksgiving and I'm probably going to put it on hold for a couple more weeks while I finish up school for the term. In the meantime, hopefully this will hold you over.

Rather than my standard discussion+problems, I will do something different today. I will first present a classic paradox in probability theory and then have a competition.

The Paradox

There are two envelopes each containing a check. The first check is for m dollars and the second is for 2m dollars. Each player then looks inside and sees how much money they have. The players are then offered the opportunity to switch envelopes.

The first player sees that he has x dollars. By not switching, his expectation is clearly x dollars. If he picked the larger of the two envelopes and switches, he will then end up with x/2 dollars after the switch. On the other hand, if he chose the smaller of the two envelopes, he will end up with 2x dollars after the switch. He then reasons that since he had a 1/2 probability of picking either envelope, his expectation from switching is

1/2*x/2 + 1/2*2x = 5x/4

Since 5x/4 > x he will profit from switching.

The paradox is that the second player has also made the same calculation and come to the same conclusion. However, both players cannot possibly profit from the switch since switching is a zero sum game.

The Competition

In honor of my stickandballguy.com email account finally working, I'm having a contest which involves anyone who is interested sending me an email there (username: greekhouse). The winner of the contest wins 500 SBG bucks and will have their name used in a future TNHP problem.

Each person who enters will send me a positive integer. Make sure you know what a positive integer is. For those of you who don't, it's a number which is both positive and an integer. Surprisingly, I did this with my Calc II students last Spring and many of them wrong down things that weren't positive integers. Most notably, people wrote down 0 and things like 0.0000000001x10^-999999999.

The person who writes down the smallest positive integer that is not written down by another person is the winner. For example, if the entries are 1,2,3, then the winner is the person who sent me 1. If the entries are 1,1, 999, the winner is the person who sent me 999.

To be honest, I'm not sure what an optimal strategy is for this game. It is possible that it depends on the number of players and might be different if there is an unknown number of players (which is the case in this instance). It is also possible that an optimal strategy doesn't exist in either case. I'd be curious to hear explanations for people's choices in addition to their numbers, so feel free to include this in the email. You may also post strategic ideas in the comments if you do not care that other people will be stealing your ideas.

The Nation Has Problems 5

GreekHouse November 13, 2007, at 6:59 pm 31 LTEs - (Sorry, but the time to add LTEs has expired)

Last week I talked about expectation with a short note at the end about sports and expectation. Whether or not you realize it, expectation is at the heart of every strategic decision that a team makes. If a team signs a big name free agent, it is in an attempt to maximize their expected number of wins (or World Series titles, or division titles or whatever). Over the course of a season, a team tries to manage its club in a way that maximizes its number of wins.

Over the course of a particular game, people have different opinions on how to maximize their chances of winning. Typically, I advocate a greedy strategy for trying to win. My goal is to basically try and score as much as possible, while conversely preventing my opponent from scoring as much as possible. There are clearly times when this strategy should be altered, but for the most part I think that you should just try to bludgeon your opponent to death.


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The Ratliff Dilemma

GreekHouse November 12, 2007, at 8:00 pm 4 LTEs - (Sorry, but the time to add LTEs has expired)

I am going to be gone this evening and so I'm delaying posting of The Nation Has Problems until tomorrow. Also, I haven't written it yet.

Instead, I thought I would pose a different kind of question to The Nation. Ratliff was included in the Garnett trade almost entirely on the basis of expiring contract. In fact, Ratliff's contract was probably necessary to complete the trade due to the NBA salary cap rules.

In the 2000-01 season, Ratliff averaged 12.4 ppg, 8.3 rpg, and 3.7 bpg—all career highs for seasons in which he played at least 50 games. He got injured in 2001-02 and only played 3 games. He rebounded and had a fantastic year in 2003-04 which earned him a monster contract with Portland. Unfortunately, he got injured again and when he was traded to the Wolves, his primary virtue was that his contract was expiring at the end of the season.

In the Wolves first couple games he looked tremendous and I posted a comment in the game thread about what the Wolves should do with him if he continues to play well. SBG commented that instead of resigning him, he might be valuable in a trade instead. He missed the next game with the flu but has come back strong in the last two games and is averaging 10.0ppg/5.5rpg/3.5bpg over 4 games and is a great compliment to Al Jeferson.

It may be jumping the gun a little bit, but if he keeps playing like this, I wonder what the Wolves should do with him. Given his history of injuries, I'd be hesitant to give him anything more than a 1-year deal. Giving him a big deal and then losing him to injury would be catastrophic to the Wolves rebuilding phase.

With that in mind, it seems like it might be a good idea to trade him. Unfortunately, due to the salary cap, no team can acquire more than 125% of the salaries that they trade away. This means that the Wolves would need to get a player (or group of players) whose salaries sum to at least $9.3 million in return for Ratliff. I'm not exactly sure how they could make that work. Any team in contention wouldn't be willing to trade another good player who is making that much. A good young player that isn't contributing to a team's playoff run wouldn't be making nearly enough.

To me, it seems like the only way to make it work would be to trade him to a team who had a good player that got injured during the season. The Wolves could then either trade for this player straight up (if he was a decent player with a few years left on the contract), or trade for somebody else who has an expiring contract plus a good, but not great prospect.

Thoughts?

The Nation Has Problems 4

GreekHouse November 5, 2007, at 6:21 pm 64 LTEs - (Sorry, but the time to add LTEs has expired)

Expectation

One of the most important concepts in sports as well as probability deals with the concept of expectation. Statistics such as ERA or points per game are attempts to use expectation to quantify how good a player or team is. Although these stats aren't perfect, they are based on the concept of expectation.

In a nutshell, the expectation of a random variable is the average value. If there are only a finite number of possible outcomes, the technique for finding the expectation is simple. We take the sums of all possible outcomes times the probability of that outcome. For instance, if we're rolling a balanced 6-sided die the expectation of the number that we roll is

1/6*1 + 1/6*2 + 1/6*3 + 1/6*4 + 1/6*5 + 1/6*6 = 3.5

This technique works even if not all the probabilities are equal. Suppose we have a coin that falls on head 75% of the time. We want to know the expected number of heads if we flip the coin once. This gives E(Heads) = 0*.25 + 1*.75 = 75%.

If we are dealing with a situation in which there are an infinite number of possible outcomes, it may still be possible to calculate expectations in this way. This would require doing an infinite summation, which unfortunately means that our expectation may also be infinite.


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Rebuilding a Franchise

GreekHouse November 1, 2007, at 8:11 pm 37 LTEs - (Sorry, but the time to add LTEs has expired)

I still remember when it hit me. I was sitting at my computer last season, looking at the Twolves payroll on hoopeshype.com and getting depressed over how terrible it looked. I had a revelation.  It was one of those moments of total clarity.  The Wolves will never win a championship with Garnett. It was a tough thing to accept and maybe that's why it took me so long. Once it happened, I was surprised it didn't occur to me sooner.

So why couldn't we win a championship with him? Simply put, the Wolves would never be able to get enough talent around him. They had a few good young players, but they were also laden with terrible contracts. Some of our young players could develop into potential all-stars, but it wasn't going to happen in time. Because of all the terrible contracts the Wolves had, there was really no hope of signing a good free agent either. By the time our young players were ready and we had some money for free agents, KG would be in the decline of his career. In the meantime, he will have wasted years of his prime on a team that never had a chance.

Once it became obvious that we couldn't win a championship with Garnett, there was only one thing we could do. We needed to trade him. Because he's one of the top players in the game we should be able to get a lot for him. The longer we wait, the less we'd be able to get. Plus, it would be in KG's best interests if we could move him to a contender. Although it might be a depressing though, it was the best option for both KG and the Timberwolves.


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The Nation Has Problems 3

GreekHouse October 29, 2007, at 7:57 pm 35 LTEs - (Sorry, but the time to add LTEs has expired)

The Monty Hall Problem

When I first announced that I would be doing this feature on a regular basis, Moss asked me a question about whether a person on the game show "Deal or No Deal" should have switched cases at the very end. I noted that it didn't matter whether or not he decided to change and then made a comparison to the famous Monty Hall problem.

I decided this would be a good topic for one of my posts, since it is famous, has an interesting and counter-intuitive solution, and is misunderstood by many people. Even the famous mathematician Paul Erdos (who is widely regarded as the best mathematician of the 20th century) allegedly got this problem wrong the first time he heard it. Part of the confusion is because the problem is ambiguously stated. Depending on the assumptions that are made, the problem can have different solutions.


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The Nation Has Problems 2

GreekHouse October 22, 2007, at 8:26 pm 42 LTEs - (Sorry, but the time to add LTEs has expired)

I had written an elaborate introduction to probability theory, but my post got to be a little bit long and I wanted to shorten it. It people are interested, I can post it later. For this post, I had a couple of interesting examples and wanted to cut to the chase.

Conditional Probabilities

Often times we want to know what the probability of something happens is given that we already have some information. For instance, we may ask "What is the probability that a team scores at least 2 runs in an inning given that we know they score at least 1?"

The formal way to generally state this question is "What is the probability of A given B?". This is given by the formula:

P(A|B)=P(A and B)/P(B)

where P(A|B) is the probability of A given B. For instance, what is the probability that the sum of two dice is 6 given that the first one is a 5? In this case A is the event that the sum of two dice is 6 and B is the probability that the first die is a 5. P(B)=1/6. What is the probability of A and B? Well, we must throw a 5 on the first die and a 1 on the second, so P(A and B)=1/6*1/6=1/36. Thus, P(A|B)=1/6.
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The Nation Has Problems 1

GreekHouse October 15, 2007, at 9:04 pm 48 LTEs - (Sorry, but the time to add LTEs has expired)

The first thing they always did was count you.

We're going to be a bit more sophisticated than the counting you did in high school, but I imagine this will be familiar to a lot of you anyway. A lot of the stuff I write about has to do with discrete, finite probabilities. In order to solve these, you typically look at how many possible favorable outcomes there are an divide them by the number of total outcomes. In order to do this, we first have to know how to count things.
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New Column!

GreekHouse October 14, 2007, at 6:02 pm 46 LTEs - (Sorry, but the time to add LTEs has expired)

The other day I posted an interesting stats question in the middle of the comments of one of SBG's posts. To be honest, the reason I posted it was because I hadn't solved it yet and I wanted to know the answer. I figured that it didn't really involve any deep theoretical math and so there was a good chance someone else would be able to get it and tell me the answer. I then promptly solved the question, but was still glad I posted it because of the interesting discussion it generated. SBG and JeffA even came up with an alternate solution to the problem that was very nice!

This got me to thinking that I would like to try and regularly post interesting problems like this. Over the course of my mathematical career I've come upon many fun and interesting problems related to spots/games/puzzles and many of them don't require a deep knowledge of math, only a bit of creativity.

So I've decided to have a regular (probably weekly) column with an interesting problem. I want to do a little bit more than just post a problem though. I want to also include a short article about some topic in math that can be useful to people in understanding the math behind a lot of the stuff I write. In particular, I would like to use my first few columns to give people the background to understand the math behind my Sharks vs. Piranhas posts. Like a lot of math, there isn't one hard trick to putting together this simulation, it is just a bunch of simple ideas put together in a useful way.
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Cribs (SBG Nation Edition)

GreekHouse September 30, 2007, at 6:38 pm 9 LTEs - (Sorry, but the time to add LTEs has expired)

I don't normally have special contacts of any kind, but today an inside source sent me some pictures of a Lake Minnetonka property that recently went on the market. Unfortunately, I am not allowed to post these pictures here, but I will give out as much information as I can.

This house is alleged to belong to a former Minnesotan who recently began work for a Boston company. The property extends south of the marker to include the tennis court and swimming pool. It also extends (I believe) across most of the peninsula. Here are some specs:

  • 12.9 acres on Lake Minnetonka
  • 32,000 Square Foot House
  • 10,000 Square Foot Guest House
  • 9 Bedrooms
  • 16 Bathrooms
  • 12-Car Garage
  • Indoor In Ground Swimming Pool
  • Outdoor In Ground Swimming Pool overlooking the lake
  • Asking Price: $53.5 million

I can tell you from the pictures I've seen that this puts Vinny Chase's House to shame. The landscaping is beautiful and the pool overlooking the lake is gorgeous.

My birthday is coming up in less than a month, so for those of you looking to get me something nice, this might be a good idea. In the meantime, I have put in a bid of $200 for it.

Better Know A Citizen – Greek House

Andrew September 11, 2007, at 10:00 am 61 LTEs - (Sorry, but the time to add LTEs has expired)

colbert.jpgName: GreekHouse
Hometown: Minneapolis, MN (although I was actually born in Fargo, ND)
Town I Currently Live In: Ames, IA
Profession: Math PhD Student
Bats: R
Throws: R
Positions: 1B,3B,DH
Greatest Career Achievement in Baseball/Softball/T-ball: When I was in college, I was nicknamed "The One Man Home Run Derby". When the fence was up on the women's softball field, I homered in over half of my at-bats. Once they took it down, my number of home runs decreased dramatically and my number of triples increased dramatically.
Hobbies: Watching sports, playing sports, playing poker, walking/running, lifting, math, movies, video games
What are you known for around the WGOM? I'm the guy who can't tell the difference between a sporting event and a math problem.
If you could have a nice, polite dinner with any 3 people - dead or alive - who and why? Leonhard Euler, Paul Erdos, and Julius Caesar. Euler and Erdos are two of the most prolific mathematicians who ever lived and I would like to get a chance to meet them and try to figure out how their minds worked. When I was an undergrad, I went on a study abroad program that was started by Erdos. Unfortunately, he had passed away a few years beforehand, otherwise I would have gotten a chance to meet him. As for Caesar, I think he lived one of the most interesting lives that any human has ever lived.
If you could pick any 3 people - again, dead or alive - to go out and party who and why? Tom Cruise, Jon Stewart, and Jesus. Just being in the presence of those three guys, I'm pretty sure every woman in the room would want to come over and talk with us and I could get the 4th hottest one.
Sports Allegiances: T'Wolves, Twins, and to a lesser extent the Vikings
Favorite Books: Harry Potter, Moneyball, Fooled by Randomness, Basketball on Paper, Various poker books
Favorite Music: Tool, KOOOOOOOOOOOOORN, Thonley, Big Wreck, Metallica, Dream Theater, Stone Sour, Slipknot, The Used, other hard rock/metal
Favorite Baseball Movie: Baseketball
Favorite Non-baseball Movie: The Big Lebowski
Favorite Food & Drink: Pizza and Lo-carb Monster
Favorite TV Show(s): Most HBO shows, The Office, The Simpsons, Futurama, 24 (although the last season sucked)
Favorite Twins Player, historically: I think most Minnesota baseball fans of my generation would be lying if they said anyone other than Pucket
Favorite Twins Player, currently: Justin Morneau
Best Twins Memories: I have great memories of the Twins 2 World Series when I was a kid. Of all the games I've been to, I'd say that Santana's 17K game is my favorite. The energy in the crowd that day was electric and is something I will never forget.
Favorite Sport to Play: Basketball, Slow Pitch Softball (with fences of course)
Favorite Sport to Watch: Baseball and Basketball are pretty close to a tie
If I could live anywhere in the world, I’d live in: I seem to be pretty happy wherever I go. I've lived in towns as small as Northfield and cities as big as Budapest and was happy in both places. Mostly, it seems to matter who the people are around me. If I had to pick one place though, I'd probably go with Las Vegas.
If I was commissioner for a day, I’d: Hug it out with Barry Bonds.
If I had Terry Ryan’s job for a day, I’d: teach Gardenhire how to use the bullpen correctly. Also, I'd challenge Tyner to a home run derby.
Favorite Ballpark: Sadly, the only major league park I've been to is the dome. I had plans to go out to SoCal this summer when the Twins were playing the Angels, but it didn't work out.
Favorite blogs: SBG is obviously the best--especially now that that GreekHouse guy is writing there. Aaron Gleeman is pretty much the only other blog I read regularly these days, although I've recently been introduced to Fire Joe Morgan and find it hilarious.
What area of math is your PhD work in? I do combinatorics and graph theory. I also have an interesting in probability and statistics, which has originated from my love of sports and gambling. For the first time since high school I'm taking a stats course this term, although I've done plenty of independent study in that area. My advisor does a lot of research with random graphs, which seems to be a good fit for me.
What would the outcome be of the "GH vs. Tyny HR Derby"? I'd like to think that my bat could back up my mouth in this case. Unlike Tyner, I am capable of hitting a baseball more than 400 feet, although I would imagine he can make solid contact a lot more consistently than me. I think it would be an entertaining affair for anyone who knows me, but pretty boring for anyone else. In the end, I will come out with a 2-0 victory.
How did the Cyclones lose to the Kent State Golden Flashes? Guh. I have no idea. It seems like every time I watch a football game of a team whose school I attend, they just end up embarrassing themselves. When I was at Minnesota, I only watched a handful of Gopher games, but everything I watched was just a humiliating loss (losing a 28
point lead in the 4th quarter against Michigan, failing to punt the ball against Wisconsin when we basically needed to get the punt off to win the game). Last year was terrible for the Cyclones and they lost every game I saw. So to answer your question, I'm just gonna say that they're in rebuilding mode.
Can we expect an encore to popular "Sharks vs. Pirhanas"? There is still a part 4 in the works. I've been pretty busy with the start of school and just haven't had the time to put it together yet. When I get a good idea like that one, I tend to ramble on for pages about it, so breaking my posts up into different parts seems to work for me. I have a few other good ideas in my head right now. I can't guarantee anything as popular as SVP (I've been overwhelmed with the response it has gotten), but these ideas will give more insight into the way I think about sports, which should be interesting to a lot of people here.

Check out the map! It's after the jump.


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Clutch Hitting

GreekHouse September 10, 2007, at 12:11 am 30 LTEs - (Sorry, but the time to add LTEs has expired)

Earlier tonight, I was debating with a friend over the existence of clutch hitting (I was of course arguing against it). I stumbled upon this article, which gives a pretty good argument saying that clutch hitting does indeed exist.  (Note: This article is a few years old.  I'm not sure if anybody has done any good follow up studies to it since then.)

The main point of the article is that if you look at the data and assume a "zero clutch" hypothesis, the data corresponds very poorly.  In other words, what has actually happened doesn't seem to correspond with what would happen if clutch hitting didn't exist.

One of the more interesting parts of the article to me was when you scroll down to the section on clutch hitters and chokers.  On the list of chokers is none other than Reggie Jackson.  That's right. "Mr. October"--the man that once hit 3 home runs in the deciding game of the World Series is the antithesis of a clutch hitter.

I have a hard time knowing exactly what to think of this.  There are two problems with people evaluating hitters as being clutch or not.  First is that they tend to base their assumptions on extremely small samples sizes (according to the article, it takes about 7 years as a regular before getting the number of PAs required).  The second is that most people use an extremely subjective definition of clutch.  Most frequently, the two are combined in order to allow people to make whatever conclusions they want about any player.

Sharks vs. Piranhas Part III

GreekHouse August 21, 2007, at 10:41 pm 10 LTEs - (Sorry, but the time to add LTEs has expired)

Hi, I'm Greek House! You may remember me from such posts as Sharks vs. Piranhas Parts I and II. I am also the author of the best selling book "Guppieball: The Art of Leaving Runners on Base".

The 3rd part of the series deals with what happens when our two teams slump. I should add at this point that I tend to think that people make too much of a big deal of slumps and hot streaks. If you're looking at a true .300 hitter and you were to pick a random sampling of 100 ABs for that player, there is about a 50% chance he will be hitting above .300 for those ABs and about a 50% chance he'll be hitting below it. If this player was almost always hitting at least .290 over 100 ABs, his true average would be way over .300.

People shouldn't be surprised when Mauer goes 0-17 or Morneau goes for a month without hitting a HR. These things may be a bit unexpected, but they're certainly within the realm of normal. These things could just as easily be attributed to bad variance as they are to an actual decrease in that player's ability.
Click here to continue reading Sharks vs. Piranhas Part III...

Sharks vs. Piranhas Part II

GreekHouse August 12, 2007, at 11:03 pm 17 LTEs - (Sorry, but the time to add LTEs has expired)

For any of you who have not yet read part I of the story, I recommend reading it first. I won't bother to reintroduct the two teams here, so you will need to at least look at part I before reading this. I had intended to include slumps in part 2, but in an effort to keep this a reasonable length, I will make that part 3 of this article. There is still plenty here for discussion!

As mentioned in part I of the article, the Sharks will win 54% of the time. This includes winning 48% of the time without going to extra innings. A little over 10% of the games will be tied at the end of 9 innings and go into extra innings. Conventional wisdom would suggest here that the Piranhas would hold a slight edge in these extra inning games due to their non-reliance on the long ball. Not so!
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Sharks vs. Piranhas Part I

GreekHouse August 9, 2007, at 9:06 pm 33 LTEs - (Sorry, but the time to add LTEs has expired)

I always find it funny when old timers talk about "small ball" and "doing the little things" as an efficient way to win baseball games. I found it even funnier when Ozzie Guillen used to talk about how his hitters needed to stop hitting so many home runs and start doing better situational hitting--as if somehow he was going to find a way to make all of his hitters hit .100 points better when there were runners on base.

The theory behind these concepts is that home runs are relatively rare and singles are much easier to come by and so through a combination of getting a lot of singles and being about to manufacture runs in close games, you will win more games. The subtly ironic thing about this way of thinking is that the exact opposite is true. Hitting a lot of home runs is not only a more efficient way to score runs, it is also more effective in scoring runs in close games.
Click here to continue reading Sharks vs. Piranhas Part I...

 
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