Student Responses on First Day Survey in Games & Gambling

I've been teaching in front of a college classroom every semester since I graduated from my undergrad institution back in 2005.  In that time, I believe I've given a first day survey every semester (except maybe my first).  The classes that I've taught have changed over the years (as has the institutions where I teach) - and the survey has changed as well, but the goal of the survey remains the same.  I want to get to know my students as people AND I want to know what they know (and what they think they know) coming into the course.

I asked the students a variety of questions on the survey - but I've only highlighted the "interesting" questions.

Why did you sign up for this course?

• Looked like a fun and informative course to fill a distribution requirement  (9 people)
• Heard it was a fun class (1 person)
• Love games (1 person)
• For distribution (without any mention of fun/interesting)  (3 people)
• Variety in math classes (3 people)

Note:  The first question was a short answer so I categorized the answers as best I could.

If you had to, which ONE of the following games would you play (with the goal to make money at a casino):  

• Craps:   1
• Roulette:  3
• Blackjack:  11
• Slot Machines:  2
• Keno: 0

The first question was designed to figure out my audience.  The second was designed to figure out what preconceived notions the students had in regards to gambling.  I also had a series of true/false and a few ranking questions on the survey (interesting to me but probably not great blogging fodder).

What information can I glean from the two selected questions?  Looking at question #1, it seems clear that the majority of the students signed up because the class sounded like a fun way to earn their mathematics distribution credit.  While some mathematicians may cringe at that idea, I think it's awesome.  It's not easy to have a non-majors mathematics class that the students are excited about before they even step foot in the classroom.

The second question was interesting to me because the majority of the class decided that Blackjack was the way to go if the goal was making money.  It's clear to me that movies like 21 (plus the glamorized depictions of casinos in movies like Ocean's 11 have an effect on people).  It was also interesting that no one chose Keno (a wise move by the way) despite the fact that Keno is also available at a lot of non-casino locations (including the Maryland state lottery).

I should mention (for those that are curious) that there isn't exactly a "correct" answer to the second question.  Keno is clearly wrong in terms of things like expected value - but if you only have $1 to bet, you have a slim chance at winning say $10,000 in Keno - unlike any other game in the list!  I didn't ask for any sort of written explanation from the students so I don't have any idea why each student chose the game they did.  The real value (for me) will be the follow-up survey at the end of the semester where I'll put the same question with a spot for a written explanation.  We'll see what they glean from the semester's worth of material!

Instant Insanity

Although one might take the title of this particular blog post to be an assessment of the craziness that is the end of another semester, I assure it's not meant to be that way.  Quite the opposite in fact.  My semester went quite well, especially the Mathematics of Games and Gambling class that I designed and taught for the first time.  As I've down throughout the semester (though admittedly not nearly as often as I would have liked), it's time for another update for the course (here's a link to my last few course updates...published back in September and another in October).

Throughout the semester, I covered a variety of topics in the class (as a reminder, the class was designed for non-mathematics majors to take as an elective to fulfill distribution requirements).  The topics that covered included (but are not limited to):

• Chuck-a-Luck
• Roulette
• Craps
• Keno
• Five Card Stud
• Texas Holdem
• Lotteries
• Instant Insanity

It is the final topic that shall be the focus of today's entry.  

For a few semesters now, I've lobbied my mathematics department to allow me to teach an experimental course.  For the longest time, I wanted to do a 300-level Introduction to Graph Theory course.  Alas, our mathematics major is quite small (and most of our majors aim to teach at secondary schools so their electives get filled by education required classes like geometry).  As such, it has been deemed unlikely that I could ever get enough students to adequately fill up a Graph Theory course.

Last spring, I changed gears and proposed the Mathematics of Games and Gambling course instead.  As you can guess, that was approved (and has, for the record, been approved for the upcoming spring semester as well).  Although I couldn't teach graph theory, I still managed to find a way to squeeze in a few days of graph theory (at a basic level) by using the 1980s puzzle "Instant Insanity."

Before I describe how I taught the lesson, I will say that overall the lesson went extremely well.  I even put a question on the final exam which almost every student got correct!  If you have taught a lesson using Instant Insanity (or something similar), I'd love to hear about it.  If you happen to be inspired in some way to use what I did in your own classroom, I'd love to hear about your experiences when they happen!

Our class periods are just over an hour long, so for the first class of the unit I showed the students the original Instant Insanity advertisement after a short PowerPoint presentation that covered the various graph vocabulary that I wanted the students to know.  

The terms weren't difficult, but they weren't all simplistic either (bipartite graphs).  

The next class was much more of a riot (for me anyhow).  I gave each student a colored sheet Instant Insanity puzzle blocks which they had to cut-out and tape together.  Watching college students do something that many haven't done since a middle-school art class was definitely amusing (and somehow a bit worrisome too)!

After constructing the blocks, the students spent the remaining time trying to solve the puzzle.  While a few did succeed, the majority of the class did not successfully complete the confounding game.  I didn't provide any hints as they left class that day, though I did encourage them to find a "mathematical way to solve the puzzle."  

The final day of the lesson was the big one - how to solve Instant Insanity puzzles using graph theory. To begin, we discussed how each block could be "unfolded" to a two dimensional image.  

From there, we can easily pair the opposite sides.

Once the three pairs are made (front/back, top/bottom, left/right), all you need is a graph with four vertices (one for each of the four possible colors).  Draw an edge that for each of the three pairings.  For example, if the top color is blue and the bottom color is red, there should be an edge connecting the blue vertex to the red one.  If the left and right colors both happen to be yellow, there would be a loop connecting yellow to itself.  In total, there should be three edges for each graph (one graph per each of the four cubes).  

Using the four graphs (or if you prefer, a single graph that combines the four small graphs), you then look for a path that uses an edge from each cube once and that makes each of the four vertices have degree two.  Once you find one such solution, find a second solution.  The first solution corresponds to the front and back of the four cubes.  The second solution corresponds to the left and right.   Note:  Once you've found the front/back and left/right, the top/bottom is forced (and therefore you do not need to worry yourself about those)!

I've skipped over a few of the finer details, but in essence that's how you solve an Instant Insanity puzzle.  For those who want to give it a go, here are three puzzles.  It should be noted that some puzzles may not be solvable.

Good luck!

The Price is Right LIVE: A Review

As I mentioned a few days ago, I took my Mathematics of Games and Gambling class to see a showing of The Price is Right LIVE at our local community arts center.  The show was last night and while I haven't seen my class yet to get their impressions of the show, here is my personal review of the Price is Right traveling show.

In a word:  Semi-lame.

Ok, that's not even a real word, but it's how I feel.

The good:

Many of people's favorite games are present.  Punch-a-Bunch, Plinko, Cliffhangers, Any Number, and Hole in One were all played.  The had one group of three people spin the big wheel.  Two people got the chance to play for the Showcase (only one showcase - both players bid at the same time).

The bad:

The show started late and ended early (or so it seemed).  Extremely short amount of time spent actually playing any games.  They showed a bunch of video clips from old shows - but nothing that you can't find on YouTube (for free).  The prizes were borderline good for most of the show...certainly not great (even with lowered expectations).  There was a refrigerator as a prize and a billiard table used as big prizes (neither were actually given away).  The small (initial bid) prizes included a 4 handheld phone system for a house (seriously, who uses those any more?), a pair of diamond earrings, a popcorn machine, and a vacuum.

The terrible:

The final showcase consisted of:
A new car (Nissan Versa I believe?)
A 3-day cruise to the Bahamas
An iPod touch
A 50 inch flat screen tv

The first contestant bid $19,000 and change.  The second contestant bid $20,000 and change.

For the traveling show, the person who is closest to the actual price (without going over) wins ONE of the items in the showcase - in this case, the 3-day cruise.  In order to win ALL of the items, you had to be within $100 of the actual price (i.e. not going to happen)!

Actual retail price (according to the show)?  $14,000 and change.

The show ended on that note.  A seemingly bogus final showcase, a pair of losing contestants, and a bunch of audience members feeling like the contestants were cheating.  The best line I overheard while leaving the theater:  "If a new car really cost $14,000, then everyone would have a new car."

Never mind all the other stuff in the showcase...

Since I was curious, here's a few numbers that I found via internet research*:
*note, all prices are guesses, I have no idea what the brands/companies were for some of the prizes
Price of 2012 Nissan Versa starting at:  $10,999 (from Nissan's website)
3-day Cruise:  $299
50 inch TV:  LG ($699.00)  one of the cheapest options
iPod touch:  $179.00
Total:  $12,176

So is the game rigged?  Well, I say yes but only because they make you think the showcase prizes are great when in reality they aren't nearly so good.  I also found it weird that in Punch-a-Bunch, the host knew exactly where the one $2500 prize was hiding...and it appeared to be printed on a larger card.  Makes me wonder if that particular hole had two cards residing in it, a $50 or similar prize if the contestant happened to select it and the big prize otherwise (the host showed the big prize to "prove" the fairness of the game).  I say when you have to "prove" that you are on the up-and-up, you probably aren't really on the up-and-up.

My suggestion to anyone who might see one of the live shows in their area - bid $1 and nothing more on the final showcase.  Chances are, your opponent will over bid and you then you win the cruise.  Don't bother trying to get too close, it won't work!

What I don't understand is why the producers of the show don't want to have one person win the cruise.  If the price is really only $300, that's paid for in a matter of 10 balcony tickets...a mere drop in the bucket.  Why have people leave angry (even if the pricing seems fair now that I looked up all the costs)?  If one of the two people had won the cruise, I think the entire audience mood at the end of the show would have been much better.

I also question the length of the show.  In a typical TV episode (granted, I'm sure footage is cut in order to fit it in 60 minutes - with commercials), there are six games played, the big wheel is spun by two groups of three people, and there are two final showcases.

In the travel show, there were only five games played, the big wheel was spun by ONE trio of contestants, and there was only one final showcase.  The entire show lasted just over an hour - and much of that time was "wasted" by showing the aforementioned video footage of old shows and for people making their way to the stage.

In the end, I'm interested to hear my students' take on the show, but for me, I can't in good conscience recommend anyone go to the show.  You'll have more fun watching old clips on your computer - save the price of the ticket.

The Price Really Is Right!

I've spent a fair bit of time discussing the happenings (and future plans) for my course that I am both developing and teaching this semester.  Today, I figured I would discuss our recent activity mostly to serve as another diary type entry for myself next semester.

My school's fall break is this week which means we get Friday off.  Yeah, a one day break isn't overly impressive (but we do get the Wednesday before Thanksgiving off as well which I appreciate).  Anyhow, it's often tough to cover much material in a short week for a distribution class since the students have all of Thursday, Friday, Saturday, and Sunday to forget.  However, thanks to our local community arts center, I got lucky.

You see, the traveling game show The Price is Right LIVE is coming to our town on Sunday.  When I learned about the show, I instantly arranged for tickets (free for the entire class) and rearranged my course schedule to accommodate the game show.  That meant both Monday and Wednesday were used to discuss Price is Right games and strategies.

Image source: http://content.clearchannel.com/cc-common/mlib/2036/01/2036_1264503009.jpg

We spent one and a half days discussing the mathematics behind the game of Plinko!  It ended up being a great way to review choice trees (using a small version of the Plinko! board).  It also natural evolved into a way to discuss using a small problem to help us solve a larger version...and it perfectly illustrated the need for a formula or shortcut when trying to analyze the "real" board.  In our case, the short cut is of course Pascal's triangle...which in itself is an excellent lead in to the next segment of the course where we will discuss the Binomial Theorem (and the MLB World Series).

After spending the better part of two classes on Plinko!, I used the remaining class time to give a crash course on some other games that might be played during the live show (I emailed the producers of the program but they wouldn't divulge which games would be played).

We looked at when to spin and when to "hold" during the Showcase Showdown.

We talked about bidding strategies.

We talked about basic strategies for a few of the games...and we ended yesterday's class with a video of what not to do which drew the expected laughs.

Overall, the students seemed quite excited for the opportunity to attend the show.  The math wasn't easy for the Plinko! board but most students seemed to intently study the various problems and explanations.  I had plenty of great questions about it during the lessons which convinced me that the students were truly engaging the material.  I believe the allure of winning a car (supposedly they play a game where someone could win a car even in the traveling show) was a great motivator!

Now that the lessons are over, I can only sit back and hope at least one of my students get the opportunity to go the stage.  It'd be a lot of fun to say that students can take my class and possibly win a vacation to Bermuda or something!

The Mathematics of Games and Gambling: A Course Update

I haven't done well keeping the blog updated on a daily basis (or even a semi-daily basis) but that doesn't mean that I haven't been keeping track of what has (and hasn't) worked in my classroom.  I've talked about my Mathematics of Games and Gambling course that I'm developing/teaching during the Fall 2012 semester a few times already (most notably here and here).  Since I haven't mentioned much about the course since then, it's probably time to reflect on how things have gone through (almost) five weeks.

So far in class (through five weeks - aka fifteen 65 minute classes), we have:

• Learned the game of Chuck-a-Luck
• Learned the game of Roulette (both American and European)
• Learned the game of Craps (both street and casino)
• Learned the game of Five Card Stud Poker
• Learned about expected values
• Learned about probabilities
• Learned about choice trees
• Learned about combinations and permutations
• plus a variety of other (smaller) topics such as a few brain teasers and card puzzles, plus assorted vocabulary as it arises in various gaming contexts.

We've also had a day where the class played craps, another where they played roulette, and today they happen to have a Five Card Stud Poker tournament.  The class has had five take home assignments so far (all but one spanning multiple classes).  Finally, the class took their first test on Monday of this week.

With all that's been done so far - and, quite honestly, that is a LOT of material for a distribution math class, it's time to reflect on the good, the bad, and the things that I might (or might not) change next time I teach the course.
*Note:  All observations are mine and mine alone.  I have not polled my students yet nor issued any sort of survey thus far in the semester.

Image source: http://singlemindedwomen.com/women-relationships/duchess-digest-the-good-the-bad-and-the-solution/

The Good:
Students love games and they seem to be working to understand the mathematics as a way to get better at the games.  I split the class into two separate tables when playing Craps - one table had every student (except one) turn their $300 into $1000+, while at the other table almost all the students went bankrupt.  It was a great, though unplanned, lesson in the draw of the casino!

The first round of testing had 6 of the 18 students earn an A on the exam.  For a distribution math course, that's awesome.  Even more impressive is the fact that the test was five pages long - and certainly not easy - and yet one-third of the class earned an A.

The best part of the class though isn't the grades or even the "fun" stuff.  Instead, the best part so far has been the students reaction to various problems, homework assignments, and in class activities.  The overall attention and engagement levels are through the roof as compared to a typical distribution class.  I can't say the class is at 100% in terms of engagement, but on any given day I would wager that at least 16 of the 18 students are fully engaged.  Again, a remarkable number considering the fact that most of the students in the class are self-professed math-phobes.

The Bad:
I mentioned the results of the first exam in "The Good" section, but I would be remiss if I didn't mention it in "The Bad" section as well.  Out of the 18 students, 4 of them ended up with Fs, including one student who left a pair of 21 point problems (out of 100 total exam points) blank.  As I said above, the engagement is there for most, but not all, students.

I think it's a bad thing that I have been unable to fit in time to play most of the games with the students and let them figure out their own strategies before we go over the mathematical analysis of each game.  I was able to do that for Roulette but not for Craps.

Things to think about changing:
I don't think I have the exact right balance of rigor and exploration.  I have a tendency to automatically analyze games logically as soon as I am introduced to them (my wife hates that, especially if we are playing against each other)!  However, just because that's a natural thought process for me, it most definitely is not for the majority of the students in the class.  I need to be better at guiding them through the analytical process.  In the beginning of the semester, I used a few brainteasers/puzzles as warm-ups which worked quite well.  Unfortunately, I haven't done much of that lately (mostly because the amount of material in the daily lessons hasn't allowed extra time).

I think the first exam should have been in week 4 rather than week 5.  I probably should have spent a full day on permutations and then a second day on combinations.  I ended up doing both on the same day and while it worked for 80% of the students, that's not close enough to 100% for my tastes.

Finally, I need to come up with a way to assess learning besides exams.  Many of the students in the class that didn't score great on the exam actually seem to know the material but they "froze" during the exam.  I think old math paranoia habits die hard...

I guess there's not really much purpose to this blog entry other than as a self-diary of sorts.  Of course, if you have any ideas for the course, by all means share them!

The Mathematics of Games and Gambling: Planning Part 2

Note:  This post originally appeared on my other (non-education related) blog Nachos Grande.  I'm slowly moving all the education related posts over to this blog.  Don't worry, there aren't many that need to be moved.  This particular entry was the second in what was supposed to be a fairly lengthy series.  Unfortunately, I never got around to writing a third planning post.  Thus, any future planning posts you see in relation to my Mathematics of Games and Gambling class will be new!

Image source:  http://entertainment.howstuffworks.com/how-to-play-roulette.htm  

This is the second in what will (hopefully) be an ongoing series throughout the summer as I plan my new course that I'm teaching this fall.  You might want to read the first post in order to see why I'm doing this and to catch up with me.

It's taken me over a month, but I finally have the first week of the course figured out and the lessons are (mostly) finished.  When I teach, I usually use PowerPoint so part of my course planning involves making the PowerPoint presentation to go along with the day's topic.  I also usually allow my students to print out the slides beforehand (if they so desire) so I have to add lots of animations to keep the work to problems (and the solutions) hidden from those who don't want to see the answers while they are working the problems out in class.  Obviously, it is a lot of work - but based on student feedback, it's appreciated by the majority of them.  Of course, there's always one or two students who tell me I should write on the board all the time instead of using the PowerPoint (they obviously haven't paid attention to my handwriting)!

Before I put together the first week of lessons, I sketched out the full schedule of the semester - and if time allows (and there's enough interest out there), I'll probably do a post covering each of the major topics as I complete them.  In this way, I can motivate myself to work harder all summer - and be accountable for my actions if I don't!

I've decided to start the course with a week long study of finite probabilities.  At the college where I work, the entire first week is known as Drop/Add week which means theoretically a student can add my class on Friday at 3:59 PM and expect to do well in the course (despite missing three classes already - it's a Monday, Wednesday, Friday morning class).  As a professor, I hate the drop/add week because you can't cover too much or new students have little hope of catching up, but you can't cover too little because then the students who attend the first week get bored and think the class is lame.  It's a tough life.

So, I usually do some form of compromise.  In this case, it's going to be a week of finite probabilities.  We'll look at dice and cards on Monday and Wednesday and then switch over to Roulette on Friday.  I think I can arrange to get a roulette wheel for the day so we'll probably "play" a little Roulette as a class as well.  I'm debating about giving each student a "gambling bank account" of fake cash to use over the course of the semester.  Will they listen to the odds and keep the money safely in the bank or will one student's success convince ten others to try their luck (with undoubtedly poor results)?  It should be fascinating (at least for me).

Once the second week begins, it's the proverbial pedal to the metal as we plow through a bunch of topics.  I think the second week will pick up with Roulette and mathematical expectation.  Speaking of that, did you know that the expected value of a $1 bet on ANY* type of Roulette bet on an American Roulette wheel is -0.0526?  That is, you should expect to lose on average 5 cents for every dollar you bet in Roulette.  No wonder the casinos love the game!

*There is one bet (the five-number bet) on the American wheel that gives the house an even greater edge of 7.89% or almost 8 cents per dollar.  See the image below for more details on the possible Roulette bets on an American wheel.

And for those that are wondering, there are two types of Roulette wheels in use around the world.  The "original" wheel (which is used basically everywhere except in the US) has the numbers 1 - 36 plus a green 0 spot.  The American wheel is identical except it adds a green 00 spot to go along with the single 0 spot.  In other words, the house advantage doubles on an American wheel!  It's not hard to see why Roulette isn't terribly popular in America and yet it is quite popular everywhere else around the world.

Image source:  http://www.bestamericanroulette.com/roulette-guide/american-roulette-odds/  

For the first week of my lecture, the main goal is for students to learn how to compute expected values (such as the one quoted above for Roulette).  Expected values aren't terribly difficult (but they prove to be an extremely valuable tool when analyzing some games)!  Essentially, you can compute expected values by summing up each individual event's probability multiplied by that event's respective payoff.  The one catch is that each of the events with probabilities have to be pairwise disjoint.

As with any course that discusses mathematical expectation, the section will conclude with a problem known as St. Petersburg paradox.  Essentially, the idea of the paradox is to flip a coin as many times as you can until you flip a tail (which forces you to stop).  The payoff is based on the number of heads you flipped before landing the tail.  If n represents the number of heads you landed, your payoff is 2^n.  The questions (that I won't answer here for those who want to think about it) are:

• What is the expected payoff for this game?  
• Is it realistic?

If all goes well, I should be able to write about games like Craps and Chuck-a-Luck and maybe even some computational rules (along with counting rules) next time as that's where the course is now headed.  I've got about two more months of summer to make this happen!

The Mathematics of Games and Gambling - Planning Part 1

Image source:  www.casinogames.org

*Note*  This post was originally published on my other blog.  I'm planning on moving my education related posts over to the new blog - what better way than to begin with the course that has inspired me to write this new blog?

By now, most of my loyal readers probably know that I teach college mathematics.  In fact, I’ve been teaching at the college level since the year I got out of undergrad (starting with a teaching assistantship in grad school).  I’ve been fortunate that I haven’t had to scramble (too much) since then in order to have a teaching job for each upcoming academic year.  For the past five years, I’ve been teaching at a four year undergraduate institution (which I love) teaching mainly Calculus and PreCalculus (though I’ve also dabbled in Combinatorics and Statistics a couple of times).  Well, I’m pleased to announce that next fall I’ll be teaching a brand new course titled “The Mathematics of Games and Gambling.”

Throughout the summer (as I plan the course), and perhaps throughout the school year as well, I plan to occasionally post a blog entry or two on my teaching progress along with any ideas/problems that I have with the course.  It is my hope that I’ll have a bit of “diary” of sorts in regards to the course.  For now, I’m thinking I’ll post the entries here at Nachos Grande but I may end up starting an “education only” blog just to keep things somewhat on theme at both blog destinations.

Luckily, for today’s post, there is a tie between one of the first topics I want to discuss in my course and the main ideas of the blog – namely, gambling!  In my course, we are going to mostly look at gambling in terms of casinos but we may also dabble in sports gambling a bit, specifically with odds and various ways to make (or lose) money.

The college that I work at is located in Pennsylvania, but we draw a lot of students from surrounding states (including a whole bunch students from New Jersey).  As such, current articles such as that discuss the legalization of sports betting in places like New Jersey will probably get a lot of play in the course.  As someone who tries to teach without always resorting to the “lecture and exam” pattern that so many math professors find themselves trapped in, I am always looking for new and exciting ways to draw my students in to the material and concepts.  Anytime you can draw in a student’s hometown (or at least home state), there is an instantaneous connection!

In fact, this particular course is designed entirely around the idea that interested and motivated students will learn mathematics!  The course isn’t for our majors (though I think there is a math major or two signed up “just for fun”), but rather, it is designed to give our distribution students a different outlet for their mathematics requirement (other than the usual statistics course).  Obviously I have no idea at this point whether or not the plan will work, but I can say that the course had strong numbers during spring registration which leads me to believe that I am on to something!

Anyhow, back to the structure of the course – one of the things that I’m debating about in my head (and now on the blog) is how much weight and importance to give various topics.  The course description is intentionally vague, and I do want to be sure to feature some games/activities that the individuals in the class find interesting, but there are other topics that I always want to make sure I include within the course. 

There will be a few topics that are no brainers – poker for one since we get a fun way to experiment with ideas of permutations and combinations.  Other no brainer topics will include roulette, black jack, and Monopoly.  I’d like to include backgammon because there’s a lot of approachable mathematics in that game, but quite honestly, I’ve never played the game myself so I don’t know that I’d be the best instructor for that one!  I plan to look at games like Chuck-a-Luck, Bingo/Keeno, and Connect Four as well.  I’d also like to discuss and investigate some online casinos, like bwin casino (which also happens to be a huge sponsor for the Portuguese soccer league cup)! 

Besides the common (and not-so-common) games in the course, I also want to spend a bit of time discussing gambling.  We will take a look at state (and multi-state) lotteries – and I have a fun little “Tips to win the Lottery” article that I printed off back when the gigantic MegaMillions lottery was all the rage.  That should draw some laughs (or at least I hope it will draw laughs by that point in the semester once students realize the follies in “hot numbers” or “numbers due”)! 

I also plan to spend a little bit of time discussing sports (and prop) betting – specifically talking about betting lines and how they affect payouts (and betting practices)!  I have already spoken to one of the psychology professors about a guest lecture on the psychology of gambling as well – it should be a great, well-rounded course!  Who knows, it may even help to shape the future direction of other distribution mathematics courses, both at my current institution and at other colleges!