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.
The first topic will likely be on the binomial coefficients (The first problem will remain a mystery for now) and will be posted tomorrow evening. Future ideas for topics include:
Basic Probability
Infinite Series
Probability distributions
Independent Probabilities
I welcome ideas for topics as well as suggestions for problems. I will give credit/post links to authors of problems when I use them, although I might wait until after the problem has been up for a while so that the temptation to google might be too great for some of you. 
These problems are not a competition and there's no prize for winning, but I will set up some ground rules anyway.
- I don't care if you cheat, but if you do, don't post the answer.
- Even if you solve the problem (or have seen it before) please don't immediately post the answer.
- Posting your solution after some time has passed is fine.
- Ask questions and state ideas about the problem to generate discussion.
- Posting solutions/hints in white text is always a good idea.
- I will post the solution eventually (assuming I know how to solve it) if nobody else does.
In a sense, all of the above are really variations of the following two ideas. I want the problem to provoke discussion and let the solution(s) evolve naturally as a result of the discussion. I don't want people to show up 15 minutes after the problem has been posted just to see the solution posted below.
Finally, I'm looking for a name for this column. I was thinking something along the lines of GH distinguished lecture series and problem of the week, but that particular title seems a bit verbose.
The WGOM: your place for half-baked solutions.
great idear, GH. In case it hasn't been made obvious already, I'm not afraid to embarrass myself in print by trying to answer technical questions.
how about "NUMB3RS"?
oh, is that one taken?
what's wrong with "Greek House"??
or
"Count Me Out with GH"
or
"Problem Child"
or
"Go Figure"
or
"Stump the Ubel"
???
These are pretty good ideas. Something with the word "Problem" in the title would be good.
Maybe "World's Greatest Online Problem".
How about "Pythagoras is in the Greek House"?
The most descriptive title from my standpoint would probably be "I don't understand this article, but it sounds interesting."
LOL
Ok, I'm obviously not going to use this, but it's funny anyway.
+10 font size, JA
I've always thought that a play on the phrase "Tools of Ignorance" would be cool.
I can't think of anything. That's why I asked for help.
Angle of Suppose
(with barest apologies to Wallace Stegner)
We just bought my daughter a copy of Math Doesn't Suck by Danica McKellar, which I thought was pretty catchy.
How about "Bert Blyleven's math does suck"?
Not Your California Math.
Danica McKellar's pretty catchy herself.
I wonder if Winnie ever thinks of Kevin anymore.
now you're drifting into porn-script territory.
I was thinking about maybe at a 25th class reunion, a well-worn and obviously drunk Winnie (who was coming off the latest in a series of bad relationships) makes a pass at Kevin who politely declines her.
I dunno. Kevin was such a dumbass, he's probably divorced now. And I'd bet he would still like a shot at some of this. (Lingerie photos, for those of you at work.)
Winnie the WHOOO!
The Nation had a discussion some time ago about Deal or No Deal. Moss saw an interesting scenario last night.
The woman got down to one case, with $750k and $1M on the board. (She set a record for highest offer as a result. Not sure if she was the highest winner or not.) She was offered $880k, which she refused.
Then she is offered the choice of switching cases. Again she refused.
Turns out, the case she stuck with was the $750k. Still a nice result, but it was the worst possible outcome from the two decisions that she made.
So Moss has a couple questions:
1) Why would she be offered more than the average expectancy value of $875k? It seems they normally offer lower than the average expectancy. And should she have taken the offer?
2) Should she have traded cases? In other words, is there any statistical difference in where the $1M is likely to be?
In my limited experience of watching the show, they typically start you off by giving a number way lower than your expectancy. As they eliminate cases, the offers get closer to your expectation. This method entices people to keep playing and makes the show more interesting. I'm not sure what benefit to the show there would be to offering her more than her expectation, except to add drama.
There is no benefit to changing unless you have some a priori knowledge about where the money is hidden (see the Monte Hall Problem). In this case, you pretty much have to assume that there is a 50/50 chance since you know they would offer you the chance to switch no matter what.
The Monte Hall problem was a central issue in the previous Nation discussions of DOND.
On the other hand, Moss wonders if truly the dealmaker doesn't know which case is which. If he does, perhaps his offer has some information in it. (Although in this particular case, if he did know, it is not readily evident from the offer.)
Even if you suppose he does know, you need to know his motives for making the offer. Is he making the offer to try and help the player because the player has currently chosen the wrong case? Or is he trying to help save the show money because the player has chosen the right case?
Because we can't even be sure of this fact, it makes it pretty impossible to infer any new information.
Unless you gather data about offers and scenarios over time.
I would guess that they offered slightly more than the expectation value because when you are that close to the end of the game, it becomes very enticing to just go for it and see if you can get the nice, round $1M. Thus, to have any chance of luring the contestant away from "going for it," you have to make a really good offer.
Unless she had an $800K debt she absolutely needed to pay off the next day, I don't think she made a bad choice. (Of course, I don't necessarily think she made a great choice, either.)
Her choice was a no-lose, since she's playing with house money and is up at least $750k.
Moss might have taken the offer, but that does remove the drama. Moss probably would have switched cases if not taking the offer, though.
The decision to take the deal (or not) here definitely creates more drama since she's left with the choice of taking a great offer or "going for it all".
Last Christmas I was in a bar with a friend of mine (who's birthday tragically falls on Xmas) and this show was on. I posed the following question to him:
"Supposed you're on a game show and you have two options. The first option is that you get a 50/50 chance of winning a million dollars (nothing if you lose). The second option is that you get N dollars. How big does N have to be before you take the second option?"
The mathematical answer of course is $500,000 since that is your expectation for the first case. The question isn't about math however, it's about risk aversion. I think most people would take $499,999 here even though it's lower than their expectation.
it’s about risk aversion
exactly. $500,000 is not the "mathematical answer" unless you define the question as "what is the expected value of this gamble?"
the utility-maximizing answer is all about risk aversion. Me, I'd have taken the $880K in less than a heartbeat.
I think it was a terrible choice. If I had $130,000 and you asked if I wanted to flip a coin with the payouts being heads - you pay me $120,000, tails - I pay you $130,000; there is no way a sane person would go for it.
I understand that she still won $750,000 but that was a given. The only thing up in the air was the additional $250k.
I think the Deal or No Deal offers when it comes down to 2 cases is always a little better than the payout expectation.
On the other hand, how many $1,000,000 winners has the show had? In some sense you're paying for a chance at making history. Given that you've already won $750,000, the marginal utility of the $5,000 in EV that you're giving up might be a relatively small price to pay.
If you have two cases left--one of them 1 million and the other 1 dollar--and they offer you $505,000, I think this becomes a very different question.
Of course, I would view it the way that you stated it. Even though I've already won $750,000, there's no way I'd pass up $5,000 in EV. If they offered me a deal of $870,000 instead, I wouldn't hesitate to gamble. Whereas in the 1 million vs 1 dollar case, if they offered me $495,000 I would take it in a second.
I wholeheartedly endorse this feature.
How does one post in white text?
<font color=white>text</font>
The easiest way I think is to use font tags. It should look something like this:
<font color="white">blah blah blah</font>
Then you get something that looks like this: (hopefully)
blah blah blah
<font color="white">like this</font>
--->testing
Thanks, gents.
How about:
This one is hilarious and self-promoting. I like it!
I say you play off of the "House" part, since House is a doctor who solves medical mysteries, and you're a House who does math mysteries.
Something like House, GH. Then it looks similar to "House, MD" and it sounds like you're James Bond. "House, Greek House"
I like this idea too, but I'd like the title to be a little bit more descriptive. It seems hard to incorporate both things into one. It could be something like "House: Math Doctor" although that just sounds way to dorky.
How Do You Solve a Problem Like Maria? Or maybe, Carry The Zero? Dull Pickle?
How about "Grecian Formulas"?
I really like that one.
+5 gray hairs
Thanks for all the great ideas guys. Right now I'm leaning towards calling it "The Nation Has Problems" or possibly "WGOM Has Problems". These are variations of the one Rhubarb came up with earlier, but are a little more inclusive of the community. "Grecian Formulas" is also very good.
If you'd like to cast a vote for a particular name, post it below in a response to this comment.
"The Nation Has Problems"
Truer words were never uttered. I cast my vote for that one.
Save "Grecian Formulas" for your forthcoming series on baseball statistical analysis
LOL...you must think I have a lot of free time on my hands.