Probability and Game Shows

[beatlefreak84]

Hi everyone,

I've finally had some time to think about other math projects I'd like to work on besides my dissertation, and one that I've had in mind for the last couple years has to do with game shows and math.

Originally, I wanted to explore a bunch of different kinds of math present in certain game shows (ex. the wagering theory for Jeopardy!, why the "Spoiler" system on Crosswords sucked, a strategy for winning Wipeout's bonus round without any knowledge of the right answers, etc.), but now, I just want to focus on probability.

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

Illinois Instant Riches:

-Mismatch (probability of winning $60,000)

What do you think?  Any other suggestions?  Any input is greatly appreciated!  

Anthony

[TLEberle]

Deal or No Deal (and even Set for Life, sadly) are rife with statistical possibilities. What are the chances of an all-blue round? What are the chances that you'll be left with (whatever) at some point?

Greed: given a question at whatever level, what are the chances that you can blindly guess the four correct answers? What if you are sure of one, and sure that one is incorrect?

Monopoly: what are the chances from space X with N rolls to go that you'll get over (or on top of) the finish line?

High Rollers: at a certain point in the bonus round, what are the chances that you'll stay alive after a roll?

[Steve McClellan]

They may be too involved for a beginning stats course, but there are a couple of interesting problems in the Showcase Showdown: the basic strategy, and the probability of each player winning (assuming all players use basic strategy). I'd love to see if someone else gets the same answers I did on the latter.

[geno57]

Blockbusters.  Are two heads really better than one?

[dale_grass]

To review the probability component of a class of mine for a test, we looked at Price clips on YooToob.  During the game, I'd pause the action and ask what the probability was of winning, getting the next number needed, spinning between $0.30 and $0.45 to win the Showcase Showdown, etc.  The probability of winning the Money Game by pulling cards at random was surprisingly high, IIRC.  Also, Plinko is just a Galton apparatus with the bottom corners cut off.  It worked great as an expected value problem.

[Marc412]

[quote name=\'geno57\' post=\'238422\' date=\'Mar 28 2010, 01:32 AM\']Blockbusters.  Are two heads really better than one?[/quote]
There's no random chance involved with BB.  Just quick reflexes and general knowledge.

Here are some other simple probability problems:
"Wheel of Fortune":  Given a random spin, what are the chances of spinning up at least, say, $600?
"Scrabble":  The challenger has to draw two tiles from 11.  What are the odds against both those tiles being among the three stoppers?
"Lingo":  What's the probability of pulling a question mark or a number that would give you a Lingo?

[Craig Karlberg]

Here's one from the most recent episode of LMaD involving the "Cash register".  What are the odds of finding not one, but TWO ZONK cards provided that the first pick is a ZONK?

[mcsittel]

Back when I was teaching intro to stats at Iowa Western CC, I found TPIR was replete with examples, so much so that you really could stick exclusively with that show (this was pre-DOND BTW).  I gathered a bunch of TPIR segments on DVD and had the class play along.  

I started with simple review things like how many possible ways there were to pick two items in Take Two or Pick-A-Pair.  We also talked about spinning $1 on the wheel in one or two spins, order of items on Hole in One, etc.  It also was helpful to talk about which of these events were truly random versus influenced by having some knowledge of the prices.  I avoided Spelling Bee because, to be honest, an intro class would struggle mightily with the details; adding the 2 "CAR" cards makes it more complicated.  Keeping it simple worked a lot better for me.

My advice would be to start with the full list of TPIR pricing games, then think about each one and what stats lesson it could provide.  Hopefully Profs. Hammett and Klauss will chime in with their ideas; I defer to their expertise in the field of teaching.

Matt (permanently on sabbatical)

[Lirodon]

Monty Hall Problem.

Enough said.

[That Don Guy]

Here's a relatively simple one:

Showcase Showdown - probability of the first person winning, if you assume that the second person would stake a second spin with less than 50 even if ahead.

-- Don

[geno57]

[quote name=\'Marc412\' post=\'238431\' date=\'Mar 28 2010, 01:52 AM\'][quote name=\'geno57\' post=\'238422\' date=\'Mar 28 2010, 01:32 AM\']Blockbusters.  Are two heads really better than one?[/quote]
There's no random chance involved with BB.  Just quick reflexes and general knowledge.
[/quote]

I dunno about that.  Wouldn't you think one of two people would be more likely to know an answer, than a single player would?  But when the show ended, the tally didn't find that to be true.  The two-player teams ended up only with a statistically insignificant leg-up.

[beatlefreak84]

Thanks for the suggestions folks!  My first thought was, "Wow...so many game shows involve essentially a variation of urn problems!"

I definitely liked the Greed/Scrabble examples provided, though, so I will add those to my list.  I've been meaning to look into Money Game on TPIR for a while, and I've wondered how much difference completely random guessing vs. knowing which three choices are the possible starting digits makes.  The latter might be a little too advanced for a beginning stats class, but the former certainly isn't.

I used to think that analyzing Spelling Bee would be much too complicated, but, actually, Steve McClellan's solution in a prior post showed me it's not too bad; it would certainly make for a decent extra credit assignment!

I thought of a couple more last night, both concerning 10 Chances:

-Show that you are guaranteed to win the small prize every time.
-Show that knowing the "ends in 0" rule guarantees you a shot at the car; how many shots are you guaranteed in this case?

In response to Blockbusters, doing a "two heads vs. one" argument is not feasible theoretically, especially since BB is not a game of chance at all.  Best you could do would be an empirical study; that is, play a bunch of simulated games, assuming each player has an equal chance of buzzing in, and see the proportion of wins.

If you can think of any more, please feel free to pass them along!  

Anthony

P.S.  I'm definitely going to keep the suggestion of using Price clips in class from Matt and Dale (I've done this with PYL with some success); thanks for that guys!

[Joe Mello]

As part of a larger stats paper, I did a section involving How Much is Enough, specifically at what point in the round was a contestant bricked.

[davidbod]

John Haigh did a book called IIRC 'Winning with Probability' that solved the Blockbusters and TPIR wheel scenarios, as well as Weakest Link banking strategy and some Millionaire analysis.

[Mr. Armadillo]

[quote name=\'geno57\' post=\'238450\' date=\'Mar 28 2010, 12:01 PM\']I dunno about that.  Wouldn't you think one of two people would be more likely to know an answer, than a single player would?  But when the show ended, the tally didn't find that to be true.  The two-player teams ended up only with a statistically insignificant leg-up.[/quote]
Remember, the the team of two had to get more answers right in order to win the game.  The fact that they were more likely to get each individual hex, but needed more hexes to win, was the balance that kept everything in check.

[quote name=\'beatlefreak84\' post=\'238454\' date=\'Mar 28 2010, 03:02 PM\']I've been meaning to look into Money Game on TPIR for a while, and I've wondered how much difference completely random guessing vs. knowing which three choices are the possible starting digits makes.  The latter might be a little too advanced for a beginning stats class, but the former certainly isn't.[/quote]
I dunno; the latter is easier to explain than you'd think.  You get a total of five guesses, and it could take you one, two, or three guesses to find the starting digits, with equal probability.  If it takes you one shot to get the front, you have four chances to get the back; if it takes you two for the front, three; and if it takes you three for the front, two.  Since each of the six possibilities for the back half are equally likely, the odds of finding it with four, three, or two guesses are (4/6), (3/6), and (2/6), respectively.  

Since each of the three are equally likely, the final probability is (1/3)(4/6) + (1/3)(3/6) + (1/3)(2/6) = (4+3+2)/(3*6) = 9/18 = 1/2.