Episode Transcript
(Intro Music Starts)
SH: Hello everyone, I’m Sam Hansen
SW: And I’m Sadie Witkowski.
SH: And you are listening to Carry the Two, a podcast from the Institute for Mathematical and Statistical Innovation aka IMSI.
SW: This is the podcast where Sam and I talk about the real world applications of mathematical and statistical research.
(Intro music ends)
SH: No preamble this time Sadie, I promised you more casino mathematics at the end of last episode and if anyone bet against me I am here to let you know, that bet won’t pay off
SW: Hahaha. Come on Sam you know none of our listeners would ever bet against you
SH: You are right there, I do know that. But do you know what I didn’t know about before I started reporting this series on the math and stats behind gambling?
SW: I’m betting quite a lot
SH: (Interrupting) Come ON
SW: (Continuing on while laughing) But I am guessing you are looking for a rather specific answer
SH: I am, in this case that answer is the game craps
SW: That’s the one from the movies where everyone is gathered around a big table with raised side and blowing on dice before throwing them?
SH: That’s the one, and that is also about all I knew about it before talking to our guest, and author of the book Mathematics of Gambling and Games, an Introduction to Mathematics, Assistant Vice President at SUNY Erie, mathematician David Taylor. Thankfully he was able to teach me quite a bit more
DT: Craps is complicated and not at the same time. So the main bets in craps are about figuring out what the shooter is able to do. So in craps, the shooter has the dice, you want them to get in a 7 or 11 on their first roll. If that happens, the pass line wins. So if you bet on the shooter to win, it's even money. You get a dollar back. if you bet a dollar.
SH: That was the non-complicated part, here comes the complicated one
DT: Then if they don't get a 7 or 11, then you're playing this other game of they're trying to roll whatever they rolled. If it's a four or five, six, eight, nine, or 10 before they hit a seven, before they crap out. So you think about this version of, okay, well now, instead of trying to get a seven, which is a very, it's, most common number you're going to roll on a pair of dice. You're trying to not get a seven, you want to hit your four again, if that's what you got on your first roll.
SW: Ok, so let me make sure I got this. If on the first roll the dice thrower gets a 7 or 11 that’s a win
SH: Correct, though David was simplifying it a bit because if the shooter rolls a 2, 3, or 12 on the first roll that’s a loss
SW: Great, more possibilities to keep track of. And, to make this even more a high school algebra problem, if they roll a value X that is not 7 or 11 they have to keep rolling until they get X again. If they do, they win
SH: Yep, but what if they roll a 7 again?
SW: They lose?
SH: You go it
SW: Ok, so from a mathematics perspective there must be a lot of probabilities going on here.
SH: There really are. That first roll the shooter has just over a 22% chance of rolling a 7 or 11 and just over 11% chance of crapping out. If it goes into the re-roll there is around a 16% chance of crapping out each roll and various chances of winning between 33 and 45% that 4, 5, 6, 8, 9, and 10 win. Overall if means that for a pass bet the probability of winning is approximately 49.3%
SW: So just about even money, and what does that make the expected value for a bet
SH: That makes the expected value of a $100 Pass bet about $98.60, or in other words a loss of $1.40 per bet
SW: So wouldn’t that mean that if you bet against the shooter that you should have a positive expected value since the probability of it happening would be greater than 50%
SH: You would think that, but the house has to win. Which is why in craps the ‘Don’t Pass’ bet does not pay out if a 12 is rolled on the first roll. A chance which brings the probability and Expected Value of Don’t Pass to the exact same as Pass
SW: Of course they do
SH: That said, there are other bets that are possible in craps, like boxcar, which rolling a twelve, or snake eyes, which is rolling a 2, which have different pay outs and different expected values
DT: Boxcar, Snake Eyes, those have anywhere from, you know, four to 10 cents house advantage. your best thing to do in craps if you want to try to make money and you want to play for a while is to bet either on the shooter or against the shooter and just play for a long time
SH: That said there is a mathematically sound strategy, it just may not make the others at the table with you very happy.
DT: I hate to say this because it's sort of bad culture at a craps table that sort of ends with the shooter but there is a slight player advantage compared to the past bed compared to betting on the shooter is betting against the shooter. It's very negligible, but if you're trying to get money, you should bet against the shooter. And whatever they'll let you do for your free odds, bet as maximum as many as you possibly can.
SW: David mentions free odds at the end there, what are those?
SH: This refers to any bets made after the initial roll, as with those bets the odds of Pass and Don’t Pass are equal with no house edge
SW: So why not just only bet then
SH: Well as we know casinos pay a lot of attention to their expected values as well, so they only allow people to bet in the re-roll section if they bet on the first roll. And they will also limit bets to a multiple of the original bet like 2 or 5 times, though some could go as high as 10 or 100
SW: Then again as David mentions there is a strong psychological element at play here that will cause people to be less likely to make the don’t pass bet. I know I wouldn’t want to be rooting against a shooter
SH: Which is why David suggests a different approach if you want to make that bet. Play the video version
DT: There you can easily sit down, play your craps game, bet against the crappy computer version, and feel better about it.
SW: But isn’t being there at a table with a bunch of other people part of the fun?
SH: I am sure it is, but are you trying to be entertained and have fun or are you trying to maximize your possible gambling return?
SW: It is still a barely negligible edge, right?
SH: Yeah
SW: Then I am in it for the entertainment
SH: Honestly, I think that’s the best choice. I want to talk to you about two more casino games where that may not be the case, but first…
SW: Another University of Chicago Podcast Network show they might want to check out
SH: You win that bet
(Ad music)
SH: If you're enjoying the discussions we're having on this show, there's another University of Chicago podcast network show you should check out. It's called Big Brains. Big Brains brings you the stories behind the pivotal scientific breakthroughs and research that are reshaping our world. Change how you see the world and keep up with the latest academic thinking with Big brains. Part of the University of Chicago Podcast Network.
(Music Ends)
SH: Up until now we have been talking about chance based casino games, so it’s time to start talking about skill games
SW: I have heard of games of chance versus skill but the difference has never been that obvious to me
SH: Well we mathematicians love definitions so let’s use the ones from the Nevada Gaming Control Board: A “Game of chance” means a game in which randomness determines all outcomes of the game as determined over a period of continuous play. And a “Game of skill” means a game in which the skill of the player, rather than chance, is the dominant factor in affecting the outcome of the game as determined over a period of continuous play.
SW: Ok, so clearly slots, roulette, and craps are all governed by randomness, no matter how good someone claims to be at rolling dice, and are therefore games of chance. What are games of skills
SH: Games like contract bridge and mahjong are examples but the big ones in a casino are blackjack and poker. Let’s start with blackjack
DT: Blackjack is probably everyone's favorite thing to think about. It's certainly romanticized in movies such as 21
SW: Of course, that was the movie about the MIT Blackjack team that used card counting and other strategies to beat the casinos. And I actually got my school to take us to see it as a math field trip
SH: Really?
SW: I’m not even kidding (laughs)
SH: That’s amazing and yes tha’s exactly what they did, and those strategies were very computational and mathematical in nature. Not that blackjack was the first casino game that was beaten by some college students
DT: But it was also, you know, if you go back and look at roulettes, if you look at a roulette wheel, there are these diamond shaped things that extend up on the roulette wheel that make the ball bounce a little bit. One of the main reasons those were added because years ago, I think this was in the 1960s, another group of college students. without that randomness, they could measure loosely how fast the ball was spinning, tap their foot, and figure out with some certainty that it's going to end in this quarter of the roulette wheel. And yeah, I mean, that's still a lot of uncertainty, but when you're paying out based on one of 38 possible outcomes and you've narrowed it down to I know which of these 10 numbers it's going to be. You can really get an advantage there. So there always is this pull and tug in any game of, here's the game, somebody figures out how to possibly break it. Somebody figures out how to adjust the game to make it computationally impossible to figure out where it was gonna land
SW: And I am guessing that after the MIT blackjack team demonstrated how successful their strategies, the casinos had to respond
DT: In the case of card counting, There are continuous shuffling machines in a lot of casinos now where card counting relies on the ability to keep track of how 10-rich is your blackjack deck. And if you always shuffle the cards after each hand, there is absolutely no advantage to counting cards. And that 3% player advantage you can get with perfect card counting, with teams and whatnot, it's just... just goes away. So it is just this pull and tug, but casinos make money.
SW: But without this card counting advantage is it just back to a game of chance?
SH: Not at all
DT: I'd say blackjack is certainly a skill-based game because it's easy to not have enough skill to lose more money.
SW: David is pulling no punches is he?
SH: He’s just sharing some hard truths. Though he has some softer ones to share too
DT: There is a basic strategy table that dealers will help you learn that you can take to the table. Even with basic strategy, the house is an advantage. They want to encourage you to feel like you have control and you're going to do well. But it's figured out based on the number of decks being played, the kind of shuffling being used, you know, what is your mathematically best thing to do, given any hand you have versus the dealer's unknown hand.
SW: That’s good. I mean that the dealer would help players with simple mathematical strategies
SH: As David says even with those the casino still has an advantage so helping the players feel like they have some control is a great way to keep them at the table
SW: True, so what sorts of things do these strategies help players with?
SH: There are a lot of books out there that people can read to learn more about mathematical strategies says about how to play blackjack but the basic ideas are around things like
DT: Do you want to hit? Do you want to stay? In some cases you can double down, split. And sometimes it's a terrible thing to do. You know, you probably don't really want to split 10 value cards because a 20 is a very strong hand. Now, aces, I mean, if you know there's a lot of 10s coming, split aces because you probably expect to get a blackjack and blackjack pays a little more, which is fun.
SW: How about poker, what does mathematics say about that one
SH: Well first of all there is an important difference for Casinos between poker and all of the other games we have discussed up to know
DT: Poker is entirely a skill-based game. Casinos will have it and the casino makes money by just taking a small piece of the pot for each hand. That's a guaranteed way for a casino to make money. No matter the hand, we're going to take five dollars from each hand and the players can play with their millions, but every hand played the casino is getting money.
SH: And since it is entirely a skills based game the way we look at it mathematically is also a bit different. There are not specific pay outs for a bet, it’s all about what the players are willing to bet on the hands they are dealt. And not only that, before we can talk about the mathematics of poker at all we have to specify the type of poker we are talking about
SW: For sure, I remember growing up it was always 5 card draw where we got 5 cards and could discard some of them and get new cards. Now, as a Texan, it’s all about texas hold em
SH: Yeah, I remember that switch as well. I am pretty sure ESPN and its airing of the world series of poker that made me have to learn how to deal with being dealt 2 cards and then making a hand with those and the 5 shared community cards of texas hold em instead of getting dealt 5 of my own cards. And since texas hold em is now the most common form of poker played it’s what David and I mostly talked about when it came to the mathematics of poker, starting with those two hole cards each player is dealt
DT: You can mathematically look at each possible pair that you have for your starting cards. You know, two, seven, not sharing a suit is traditionally considered the worst hand. because you're not long enough to get a straight. You can't get a flush. So you're really relying on, is there a possibility you're gonna get a pair of twos or a pair of sevens, maybe a full house, but if anybody has eights or higher for a pair or part of their full house, they're gonna beat you.
SH: And while the mathematical strength of a hand is important it not the only thing to consider when analyzing a hand of poker
DT: the sociology of the play the psychology of what's going through each player's mind do you bluff do you not bluff do you get that two seven offsuit and decide this is the hand i'm going to bluff i have a pair of kings and i'm going to pretend that i have a pair of kings and i'm going to play as though i have a pair of kings. It works some of the time, versus players that know what they're doing it doesn't work most of the time.
SW: Because players who know what they’re doing can really read people who bluff…
SH: That is certainly a possibility, but mostly players who know what they are doing have studied the probabilities of hands and know what is possible given the cards on the table and they know exactly how strong their hand is. Which means
DT: In a lot of cases, it is the stronger hand that you can play to a better outcome.
SW: So bluffing is a losing strategy?
SH: Not all the time, actually you can even figure out mathematically when it is the optimal choice
DT: You can figure out, should you bluff or not? based on how much you expect to lose, how strong your hand is. I don't think anybody does that live. I think they get a feeling as they learn and play the game. But one situation I did in my textbook that I wrote is, "Figured out, you know, just a base. Here's the assumption. I've got $50 involved, the pot's $100. This is what I need to do. Should I bluff or not?" The game theory outcome was I should bluff about 5% of the time.
SH: Which is why David says
DT: When you see pros play poker, they don't bluff a lot. They're playing the hand they have
SH: Then again poker does have yet another variable it can introduce to make things even more complicated, the number of people at a table. And when that number gets small, especially when it is just two, all sorts of things like having to ante for a hand before it is dealt can introduce wrinkles into those game theoretic optimums
DT: Final two, there's a lot of, I mean, there's the forced betting, you know, hey, I'm half in, you're hole in, I don't even want to play, I give, I fold, or I'll call your minimum bet, like, yeah, I'll call two and we'll just see what the cards are.
SW: So in the end if you want to win at poker you need to do more than just study probabilities?
SH: 100% yes. Probability, statistics, and game theory would give you a great mathematical base, but math and statistics alone will never be enough to make one great at poker
SW: How about neuroscience?
SH: I got a deck of cards right here, let’s see
SW: (Laughs)
(outro music)
SH: If you or someone you know is struggling with a gambling problem, help is available. The National Council on Problem Gambling provides a range of resources, including the National Problem Gambling Helpline™ You can reach them at (1-800-MY-RESET) to help connect you with local resources.
SH: Don’t forget to check out our show notes in the podcast description for more about David’s work and a link to his book
SW: And if you like the show, give us a review on apple podcast or spotify or wherever you listen. By rating and reviewing the show, you really help us spread the word about Carry the Two so that other listeners can discover us.
SH: And for more on the math research being shared at IMSI, be sure to check us out online at our homepage: IMSI dot institute. We’re also on Bluesky and instagram at IMSI dot institute, as well as instagram at, guess what, IMSI dot institute! That’s IMSI, spelled I M S I.
SW: And do you have a burning math question? Maybe you have an idea for a story on how mathematics and statistics connect with the world around us. Send us an email with your idea!
SH: You can send your feedback, ideas, and more to sam AT IMSI dot institute. That’s S A M at I M S I dot institute.
SW: We’d also like to thank Blue Dot Sessions for the music we use in Carry the Two.
SH: Lastly, Carry the Two is made possible by the Institute for Mathematical and Statistical Innovation, located on the gorgeous campus of the University of Chicago. We are supported by the US National Science Foundation and the University of Chicago.
SW: Buh Bye
SH: What is ringing? Oh, it’s my computer, I think I am just hearing it in my headphones
SW: Haha. Yeah, I hear nothing
SH: No, no it’s coming out of the speaker
SH: I need to put down this mini screwdriver
SW: (Laughing) It’s not even a pen it’s a screwdriver
(Music ends)
SH: (Vocalizing a stead note) I think I might just replace all of our music with pure sine tones
SW: People will think it’s like the warning sirens on campus
