Probability and Game Shows

[davidhammett]

[quote name=\'beatlefreak84\' post=\'238406\' date=\'Mar 28 2010, 12:25 AM\']Specifically, I'd like to collect some probability exercises based on game shows that would be suitable for a beginning statistics course.  Thanks to some old posts here, and some work on my own, I've come up with a few, but I'd like more if anyone has any, or even just some suggestions on what might be feasible to work on.  If I get a good amount, I'll try to turn it into a paper in the near future.
Anyway, here's what I have so far; note some of these are more substantial than others:
Price is Right:
-Let 'em Roll (probability of winning a car)
-Spelling Bee (probability of winning a car and probability of a "perfect game")
-Pass the Buck (probability of winning a car and probability of winning at least something)
-Five Price Tags (probability of winning given completely random guesses on "True/False" game)
-Secret X (probability of winning given random distribution of X in the center)
Press Your Luck:
-probability of hitting a Whammy on any given spin (assume board is completely random)
-showing Larson's run is statistically significant[/quote]
Pardon the lengthy response, but I've been away for a few days and wanted to address what I'd read so far.

Anthony, I think the key here is *beginning* statistics course.  All of the above examples seem reasonable, although not necessarily easy, for a beginning course.  The exception would be the probability of hitting a Whammy, which is trivial to determine once you know the layout of the board.  An alternative there might be to have students compute the probabilities of making N spins in a row without hitting a Whammy, which is useful information to have when deciding under what conditions to use or pass those remaining spins.  (You could then work in some sort of expected value calculations if you wanted.)  I've never thought of showing Larson's run was statistically significant, but that sounds like a great exercise!

I like Travis' suggestions as well.  Regarding Greed, I got a call from Bob Boden early on asking essentially that same question, and thanks to some quick recall of Pascal's Triangle I got my job helping to develop the show.  Needless to say, it's an example I use in class now any time I want to illustrate a real-world use of combinatorics!  Also, in playing High Rollers with a class, I've asked just those kinds of probability questions that Travis suggests.  And the Monopoly bonus round offers a great opportunity to introduce or illustrate the technique of Monte Carlo simulation to solve a problem.  

Steve's Showcase Showdown example is another opportunity to use simulation, or for that matter it can be done deterministically; what is interesting is to compare whatever results you get (assuming a "random" wheel) to the empirical results you might get from several episodes of the show.  My understanding is that the comparison shows that the wheel is clearly NOT random -- not because it's in any way rigged, of course, but rather because there are limitations into how strongly most people can spin the wheel, and also because the wheel always starts in the same place at the beginning of every Showdown.

Indeed, TPIR is full of opportunities for analysis; one of my current students is figuring out probabilities involving "Spelling Bee" and "Any Number" as part of his project for the class.  I also give my classes the exercise of doing Plinko expected values during one of our "fun" days after the AP exam is done.  And as Marc suggests, Lingo offers some good opportunities as well, as I know first-hand from having worked on its development as well.  (For an advanced exercise, try to compute the probability of someone winning the end game if they can draw up to N balls.)

I agree whole-heartedly with Matt's assessment of some of the TPIR games; that's why I only assigned Spelling Bee to a student that I felt could handle all of the cases.  (He's also a game show fan, which certainly helps!)  And the Monty Hall problem is great for derailing a class for up to half an hour while you discuss the many different ways of figuring out the problem; I always like to point out that the Barker's Markers game on TPIR contained some of the same elements in its analysis.

Anthony, if you haven't already done so, check around online for some other sites that may already have some of this material ready for you to use.  One that I've found is http://statweb.calpoly.edu/mcarlton/gameshows/index.html, and I'm sure there are others.  Good luck!

[beatlefreak84]

Thanks for all the info and suggestions, David!  I took a look at the website you mentioned, and was quite impressed by the amount of material there, especially on Plinko.  Any of those would make for fine activities, and I'm glad others have thought about this as well.

For the PYL problem (of statistical significance), the one I typically do in class is of the same variety, but I use the famous "Spin War" episode between Lori and Cathy since I can only show the second round and get enough data.  Unfortunately, the only class I've taught using stats teaches it along with a host of other topics, which makes the course fun, but really limits what I can do as far as activities go since there's only so much time.  In the future, however, I may use the lovely YouTube to allow people to watch the Larson episode at home and work the problem there.

So, I'm hoping that this project will allow me to live vicariously through other stats teachers...

Anthony

[Joe Mello]

Of all the PYL examples, the best I would think would involve the sentence "You have one spin left."  You could easily make a full page on exam based on that single scenario.

On a particular game show...(explain show and board)...You are in control and have one spin remaining.  Assume you are currently indifferent to spinning, that is, the chance you are spinning or passing is equally likely.  Also assume that the chance of hitting any value is equal.  What is the probability that after the spin has been taken:

• You win
• You lose
• You don't win
• You don't lose
• The game doesn't end
• You take the spin and...
• You pass the spin and...

[TLEberle]

[quote name=\'beatlefreak84\' post=\'238781\' date=\'Apr 2 2010, 04:06 PM\']So, I'm hoping that this project will allow me to live vicariously through other stats teachers...[/quote]What level do you teach, Anthony? If your class has the basic fundamentals of math down pat, the sky's the limit. If not, then you sorta have to dial it back. And if they have the skills, there's no reason to go nuts: the combinatorics of Greed or Tic Tac Dough, the "inverse" property being used to solve the odds of winning a top prize at the Punch Board, or the chance of the top prize NOT being spun up on Wheel of Fortune in X days.

[beatlefreak84]

[quote name=\'TLEberle\' post=\'238877\' date=\'Apr 4 2010, 11:35 PM\'][quote name=\'beatlefreak84\' post=\'238781\' date=\'Apr 2 2010, 04:06 PM\']So, I'm hoping that this project will allow me to live vicariously through other stats teachers...[/quote]What level do you teach, Anthony? If your class has the basic fundamentals of math down pat, the sky's the limit. If not, then you sorta have to dial it back. And if they have the skills, there's no reason to go nuts: the combinatorics of Greed or Tic Tac Dough, the "inverse" property being used to solve the odds of winning a top prize at the Punch Board, or the chance of the top prize NOT being spun up on Wheel of Fortune in X days.
[/quote]

I teach college-level, but usually not statistics (I'm teaching differential equations this semester, for example).  The course that I've taught involving stats is called Contemporary Math, which is essentially a "Math for Liberal Arts Majors" course.  So, many of my students only have basic algebra/geometry skills from high school, and, surprisingly, a good number of them have no experience with probability.

The probability part of the course only lasts about 2.5 weeks, so it severely limits what I can bring into class (2.5 weeks sounds like a lot, but there's a lot of material to get through).  Nevertheless, I try to bring in a good number of game show examples just to break up the glut of casino game examples.  This actually prompted one student, on an assignment where I asked them to write a "tip sheet" for anyone who would be taking the course from me in the future, actually gave as a tip, "Learn to like game shows."

The two mainstays I always use are the Monty Hall Problem (and then ask them why the DOND case-switch offer at the end is not the same thing) and the PYL problem I mentioned previously.  I definitely want to bring in more TPIR examples, though, especially since I always end up teaching this class when TPIR comes on.  For some reason, I just never get around to it.

Anthony

[TLEberle]

[quote name=\'beatlefreak84\' post=\'238878\' date=\'Apr 4 2010, 10:50 PM\']I teach college-level, but usually not statistics (I'm teaching differential equations this semester, for example).  The course that I've taught involving stats is called Contemporary Math, which is essentially a "Math for Liberal Arts Majors" course.  So, many of my students only have basic algebra/geometry skills from high school, and, surprisingly, a good number of them have no experience with probability.[/quote] I know all about this, because when I was in high school it was called Special Topics, and I used it to buffer the craptacularisticy grades I got in Algebra and Computer Programming.

The probability part of the course only lasts about 2.5 weeks, so it severely limits what I can bring into class (2.5 weeks sounds like a lot, but there's a lot of material to get through).  Nevertheless, I try to bring in a good number of game show examples just to break up the glut of casino game examples.

This might be more suited for psychology, but I think showing the house take on particular games is great. If they choose to punt their money on Powerball or the Pick Three, at least they're informed.

This actually prompted one student, on an assignment where I asked them to write a "tip sheet" for anyone who would be taking the course from me in the future, actually gave as a tip, "Learn to like game shows."

It isn't like any of these examples are particularly esoteric (like when I spent a while with the Encyclopedia and my TI-86 trying to compute the odds of someone winning the $250,000 advertised prize on some 1950s-ish game show or other.

T$R put out a "how well do you know your friends" game, and one of the questions was "You have won $20,000 on a game show and can bet it on the Triple Your Money wheel; 50/50 that you get paid or go home poor. Would you do it?" This is an easy way to calculate things like expected value of an event, but also to generate discussion of why someone as risk averse as me would grab the $20,000 in the briefcase and run like hell even though the payoff is more than the risk.

[MikeK]

[quote name=\'beatlefreak84\' post=\'238878\' date=\'Apr 5 2010, 01:50 AM\'](I'm teaching differential equations this semester, for example)[/quote]
I'm sorry.  Please accept my condolences.  It's been 7 years since I took Diff. Eq. and I still have the occasional nightmare over that class...

I have done several experiments and lessons, and given projects based on game show probability.  When Deal or No Deal first became popular 4 years ago, I had honors-level juniors and seniors determine the number of ways cases could be picked (essentially xCy, where x is the number of cases available and y is the number of cases to be picked), then asking them to find various probabilities--picking the top 2 values from 8 cases, picking the top 3 values from 11, etc.  As a follow-up/extra challenge, I let the students loose on the UK 22 case version and their 5-3-3-3-3-3 method of case selecting.  With my lower-achieving 9th graders, I keep the tasks much simpler--basic probabilities from a variety of pricing games, from Wheel of Fortune, and Deal.  Somewhat to my surprise, TPiR is almost a universal language.  When I've referred to it in class, students almost always ask me "What happened to that old guy?"

One situation I've been pondering, even though it has to do more with game theory and psychology than probability, is the optimal way to play Pot of Gold from Illinois Instant Riches.

[davidhammett]

[quote name=\'MikeK\' post=\'238884\' date=\'Apr 5 2010, 09:31 AM\']One situation I've been pondering, even though it has to do more with game theory and psychology than probability, is the optimal way to play Pot of Gold from Illinois Instant Riches.[/quote]
Thanks, Mike; I was blanking on the name of that.  But yes, that game is at once both simple and incredibly complex to analyze, for just the reasons you mention.  I haven't seen any reasonable treatment of it; let me know if you come across something or work on it yourself!