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by Paul Wilson

BJI contributing writer Paul Wilson is a quasi-Renaissance man and graduate of Millsaps College. Some of his interests and hobbies include finance, consulting, travel, photography, and rock music. He's an avid baseball fan. Paul has done freelance writing and editing for gaming publications and takes blackjack, video poker, and sports betting very seriously. As we learned in the November 2014 issue, he also might have a "thing" for Wonder Woman.

I'm often asked questions about casinos, blackjack, sports betting, video poker, and even casino promotions. Inevitably it seems my answer generally begins something like this. "It's not that difficult. It's just high school math." In this month's Paul's Pointers, I'm going to talk about numbers, casino math, and what it all means to you.


You've probably heard the phrase "the house always wins." As much as I hate to admit that, it's pretty much true in the long-term. It doesn't have to be on the backs of you or me though. There are plenty of players in the casino universe for the house to prey upon. So how does the casino consistently beat its players? It's something called the house edge.

All casino games, with the exception of a few video poker pay schedules, are designed with a house advantage. Mathematically speaking, this is a measure of how much the house expects to win on each wager. It is expressed in terms of a percentage of each wager. However, I find this value to be much more useful when expressed in terms how much money a player can expect to lose (or win if they are playing with an advantage) over the course of a series of wagers and the amount bet. Let's look at an example. If the house has a 5 percent advantage on a wager, the house will win, on average over time, $5 for every $100 wagered; $50 for every $1,000 wagered; $500 for every $10,000 wagered and so on. What this means for you is that you can expect to lose $5 for every $100 you wager in this example. If the house edge were reduced to 3 percent, you would expect to lose $3 for every $100 wagered.

There are games on the casino floor where a player's strategy decisions and skill level can impact the house edge. As you may have guessed, blackjack and video poker are two prominent and popular examples. I'll talk more about blackjack math in a section below. Realize for now that a skilled blackjack player can decrease the house's expected win and a poor player can actually increase the house's expected win. The same holds true for video poker. The math has been derived on the "best possible" outcome in the long-term for each pay table, but this number will decrease with each player decision that isn't "computer perfect." For example, 9/6 Jacks or Better where the full house pays 9 coins and the flush pays 6 coins per coin played returns 99.54% with perfect play. That means that with perfect play over the long-term players can expect to lose $4.60 for every $1,000 wagered. Over the long-term, strategy mistakes will reduce the players expected return and increase the house's profit over that player. This is pretty simple stuff, right?

Two terms the casino folks use to compare their actual results to their expected results are "handle" and "hold." The handle is the amount of money that players buy in for over a specified period of time. The hold refers to how much money the casino keeps or "holds" at the end of a specified period of time. Let's say you and I sit down together at a blackjack table and each buy in for $300. We give the casino $600 in cash in exchange for $600 worth of playing chips. We've contributed $600 to the casino's handle. If at the end of our playing session, one of us cashes out with $290 and the other with $210, combined we have $500. We take our chips to the casino cashier and return them in exchange for $500 in cash. The casino hold was $100 or 83% (500 divided by 600) - a good result for the house, but not for us. As you can see, if the handle increases, then the hold should increase as well. The hold percentage doesn't have to increase in percentage terms to increase profits. Would you rather have 10% of $1 million or 5% of $5 million? Think about it. Too many casino decision-makers don't seem too.


In the previous section we learned about the house edge and what it means in percentage terms. Now let's take the math a bit further and look at how this applies to the game of blackjack since I am writing this piece for Blackjack Insider. By playing basic strategy a smart player can reduce the house's edge to about 0.5%. That means their expected loss is about $5 for every $1,000 wagered, all things being equal.

Most casinos use the concept of the "average player" when doing computations. I'd guess some casinos use different values, but most will be in the 1% to 2% range. If your play is below "average" you can be expected to lose more than 2% over the long-term and the casino will probably welcome you with their proverbial arms outstretched. If you or an "average" player, most places will tolerate you and let the house edge slowly grind away at your bankroll and make their profit over you more slowly. If you are a skilled player and consistently beat the house, don't expect much in terms of comps or invites to special events, and you may even be asked to stop playing blackjack at their casino altogether.

By now you should understand the house edge concept and what Stu D Hoss, myself, and other gambling writers mean when they say the house edge against a basic strategy player is 0.4% for example. However, do you know how they derive that value? This is the part of class where rules and getting paid properly on your blackjacks matter.

Blackjacks should pay 3:2. If they don't the mathematical construct of basic strategy starts to break down rather quickly. When a casino only pays 6:5 on player blackjacks, it increases its advantage over even the best players by about 1.4%. That may not seem like much initially. However, think about what we've learned thus far. That equates to $1.40 for every $100 bet. That could add up rather quickly and does. Let's take a look at the math. The probability of being dealt a blackjack is slightly less than 5%; approximately one every 20.7 hands. On average, you can expect to play about 60 hands per hour (this number may vary based on number of players at the table, hand shuffling of the cards versus automatic or continuous shufflers, and dealer speed). Mathematically you can expect to average about three blackjacks per hour. Let's assume you bet $10 per hand. A blackjack will return $15 at 3:2 payoff odds. If you are only getting paid 6:5 or $6 for every $5 bet, you'll receive $12 for each winning blackjack. This $3 difference per occurrence can add up. With an expectation of three blackjacks per hour, you're costing yourself $9 per hour on a $10 game ($3 difference x 3 blackjacks per hour). At higher limits, the cost to the player is even greater.

Different house rules affect the casino's advantage, positively or negatively. Variations of house rules and the introduction of more decks to the game have mathematical values assigned to them. By taking the sum of these values, one can determine the house edge on a specific blackjack game. Single-deck blackjack is essentially "ground zero" when it comes to learning and computing these numbers. By simply adding a second deck of cards to a previously single-deck game, the house edge increases from no advantage to 0.35%. This edge increases to 0.58% on a six-deck game.

Some rules are good for the player and some for the house. Let me share a couple of common ones you'll likely see in any casino these days. When dealers hit soft 17, as opposed to standing on all 17s, the house edge against the player is increased by 0.20%. The ability to double down after pair splitting (DAS) decreases the house edge by 0.13%. The ability to re-split Aces to form up to four hands decreases the house edge by 0.05%.

Say we have a six-deck shoe game and re-splitting of Aces is allowed; blackjacks pay 3:2; and doubling down after splitting is allowed. What's the house edge? With a shout out to the punk rock band the Violent Fems, let's add it up! We start at zero. Then add 0.58% for the six-deck game. Now the sum is 0.58%. We can now decrease this value by 0.13% because DAS is allowed. The sum is now 0.45%. Finally, let's account for the ability to re-split Aces and subtract another 0.05%. That gives us a sum of 0.40% (0 +0.58 - 0.13 - 0.05 = 0.40). The house edge against perfect basic strategy is 0.4%. You remember adding and subtracting decimals from math class don't you? We just did it!


I don't know what your high school math classes were like. For me it was pretty lame after ninth grade, but that was due largely in part to having a couple of poor to mediocre math teachers for geometry, Algebra II, what was called Senior Math. However, I did manage to pick up some probability and enough statistics to bluff my way through a couple of research courses in my undergrad and master's programs. So I'd be lying to say I didn't learn anything. However, what if your math teacher could answer the question above? Ours couldn't and without fail hardly a day went by my junior or senior year when one of us didn't raise a hand and ask that question when called upon. In the sections above we've used simple math to explain the house edge and delve into why blackjack rules matter and what it can mean to your bottom line. Now let's apply "high school math" to some other common casino situations you encounter in "real life."

Since it's football season and the MLB playoffs begin next week, as well as, the National Hockey League regular season, let's start in the sport book. The majority of sports bets involving a point spread are listed at -110. This means that you have to bet $11 to win $10. This is the simple reason most causal bettors can't beat the book. At 11/10 the house has a 4.5% edge. That's pretty stout. Over time you will get destroyed if you win half your bets at those odds! Let me give you an example from the world of finance to illustrate how strong 4.5% really is. According to the "Rule of 72" your money would double in 16 years at 4.5% annually if left untouched. Not flashy, but solid, especially if inflation remains in the 2% to 2.5% range. Simple math applied to the sports book and financial planning in the same paragraph; gotta love it!

Another important "real life" concept in the sports book involves math with an assist from your English teacher. If you play parlay cards or discuss odds period, know and understand the difference in TO and FOR. A winning three-team point-spread parlay bet off the board pays 6 to 1. That means each dollar you bet returns $6, plus your original wager. When you cash your $5 wager you should be paid $35; the $5 for your original wager in this example, plus $6 for each of those dollar (5 + 30 = 35). Realize the true odds in this example are actually 8 to 1 and though parlays offer the chance to win a lot for a little, they are statistically bad bets. Money line bets are an entirely different animal and pay off at true odds. Most of the printed parlay cards you'll find in Las Vegas offer payouts in the form of FOR; for example three teams pays 6 for 1; four teams pays 10 for 1, etc. This mean your $5 three-team parlay will return only $30; one dollar for each dollar wagered. Your profit will be $25 in this example versus the $30 profit in the earlier example. The 6 for 1 payoff is equivalent to 5 to 1. English class and math class in the sport book; who would've guessed it?!

Simple math hits video poker players two ways. The first is in the form of the published pay tables. If you aren't paying the highest paying version of a game, then you are "losing" more than you should. It's not uncommon to see Jacks or Better offered on today's casino floor with pay schedules that pay 8 for a full house and 5 for a flush per coin played instead of the customary 9 and 6. Each one-credit reduction from the full-pay version reduces the expected return by about 1.1%. The 9/6 Jacks or Better version returns about 99.5%. By reducing it to 8/5 the expected return drops to about 97.3%. That's an expected loss of $27 per $1,000 in coin-in (CI) versus a $5 loss on $1,000 in CI. Second, like in blackjack, you have to learn the best playing decisions or strategy for each hand. Fail to play correctly and the expected long-term returns will be lessened. Realize that these numbers are based on the long-term and anything can happen in the short-term. However, the long-term is comprised of lots of short-term events. Pay schedules and strategy matter!

Another example of "high school" math in the casino is figuring out player's club points. Most casinos offer some sort of loyalty program where your play is tracked and you are given certain perks based on your play. Slots, video poker, and at some casinos sports bets and table games play is converted into points. Often these points can be redeemed for comps or free play. It's important to compare slot clubs and the value of your play in terms of points.

Here is real-life example from a large casino chain in Las Vegas. This casino chain offers 0.1% cash back (CB) or comps. That mean each $1,000 in coin in (CI) earns $1. However, by simply swiping your player's card you get a 3x points multiplier. The same casino chain often has 6x point multiplier days. Let's take a look at this and apply it to our 9/6 Jacks or Better video poker expected returns. First our $1,000 in CI earns $3 in CB; $6 on 6x points multiplier days. If we apply these numbers to our 99.5% expected return game we can expect our long-term results for each $1,000 in CI to range from -$5 (no slot club points), to -$4 (at 0.1), to -$2 (at 0.3), to +$1 (at 0.6). By playing on 6x bonus point multiplier days, we've turned a negative expectation game into a positive expectation game. You won't get rich with small margins like this, but it adds up over the long run. If you want to stay in the game and maximize your time in the casino and your bankroll, you need to understand these types of computations and use them to plan your playing sessions.

Yet another large casino chain offers 0.1%, 0.2%, and 0.3% cash back or comps based on your player's club status. The same casinos offer a 0.40% discount at their casino-owned restaurants when paying with points. Do you want to redeem those points for free play where you are mathematically an underdog, for cash back, use them to pay for meals, or a combination of those options? By understanding the math involved you can determine that your points at those casinos have more value when paying for food. However, you can only eat so many meals, so a combination might be the best use of those points for your circumstance. At least you can figure out your options and put them into numerical (as in dollar) terms now!


I'll conclude by listing some "walking around" house advantage numbers. You can use these to determine what games you might want to learn and what your predicted returns will be over the course of many sessions. I encourage you to keep records of your gambling forays and compare every now and then. You might be surprised by your results.

In Baccarat the house edge is 1.1% to 1.2% for Player/Banker bets and 14.4% on the Tie. The Pass/Don't Pass bet in craps has a house edge of 1.4%. For prop bets it ranges from 10% to 16.7%. In single-zero Roulette every bet carries a 2.7% house advantage. This number increases to 5.4% on double-zero games. Slot machine returns vary for a number of reasons based on jurisdiction, gaming regulations, and what the casino and slot maker agreed upon before putting the machine on the casino floor. Some guidelines are that nickel slots have a house advantage of 7% to 12%; quarter slots 5% to 10%; and dollar slots 2.5% to 6%. The house keeps 25% to 30% of the money wagered on live keno. I've mentioned blackjack, video poker (varies by game and pay table), and sports above.


In this month's Paul's Pointers, I tried to explain how easy casino math really is and how you can use it to understand what you're getting into and plan accordingly. I discussed the house edge and gave numerical examples of this edge in many traditional casino games. For the most part, "carnival games" and blackjack side bets carry higher house edges than those games listed above. I introduced the terms "handle" and "hold" to explain how casinos measure table game profit margins. The significance of playing rules in blackjack was introduced and I broke down how to determine the house edge of a particular game. Minor rules changes matter and can affect your results, both in the short- and long-term. I also flashed back to high school and tried to partially answer the question "How are we gonna use this stuff in real life?"

For whatever reason, too many people in our country seem to have an aversion to math. I failed to see the connection between math and real life as a kid, but now I see math everywhere and use it every day. I'm really not any good at higher level math, but when it comes to applied math, I get it now. Paying your bills, balancing your check book, figuring out loans, credit card interest, financial planning, following sports, analyzing data at your job is all high school math if you think about it. Pretty much everything is measureable and quantifiable. Let's face it, folks; math is all around us and life is better if you make math your friend!

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