The Mathematics of the Casino | What people get wrong about gambling 

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Want to know the secret for leaving a casino with a small fortune? Arrive with a large fortune.

I am a retired casino dealer living in Las Vegas and I can attest to what he is saying. Take your shot, and win, lose or draw leave. The longer you play the more chance you have of losing. I have 36 years of experience watching this happen.

So the moral of the story is, “If the odds are against you, don’t give them time to even out”

A couple of weeks ago, I walked into a casino for the first time in my life with only $5. On my very first round of the first game I played which was a slot machine, I ended up winning $107. I instantly cashed out and walked out.

I like to factor in the free drinks if you sit at a black jack table for a while.

Probabilities are always a fascinating subject to me.. i like the way it can predict what may happen in future just using some numbers and some expressions

When I went to a casino and did this I used the Alice strategy. I started with $10, betting $5 at a time. I made about $80 before going all in a la Bob's strategy, lost it all in one go, and then lost $60 more trying to make it back. Just my personal experience, though.

This makes me want to watch the movie 21 again. I recommend buying the book called "Beat the dealer" by Edward Thorp. Goes into so many strategies in statistics and probability. Great video Zach.

I'm happy to say this channel had a part, even if small since my decision was already mostly made on pursuing engineering. I'm now a junior 1 and looking to take abstract algebra next year. Amazing channel Zach! :)

Worked for an online casino software provider for 14 years. Taught the basics of real odds vs offered odds (the difference is known as the house edge) and the only strategy to win at an odds based casino game is to own the game and encourage as much play as possible. This is why casions balk at large, single plays, and encourage the players to increase their number of plays with smaller denomination machines and bets. As game play tends to infinity, the house edge tends to a guaranteed income.

Thank you. RU-vid is flooded with "systems" to beat these games, but the basic math ALWAYS says "no." Yet, the comments sections are filled with people in huge denial, often citing anecdotes about how they beat the odds and won. If they did beat the odds, it's due to variance. We don't lose exactly on the trend line. Sometimes we come out ahead. More times we come out behind. And there's no way to predict when that is. If a casino game could be beaten, they'd change they payouts to make it less profitable so players would still come out behind. This happens with video poker, where the payout table is crucial. (And usually ignored by players, to their detriment.) The only casino game that can be beaten is blackjack. But to beat it, you have to play perfect strategy, maintain a running count, convert that to the true count, play variances on perfect strategy based on the count, and vary your bets widely when the count is in your favor. And you have to do all of this while not getting detected by the casino staff. Contrary to popular belief, counting cards is not cheating, nor is it illegal. But the casino can "back off" a player it suspects of card counting, refusing to take his/her bets. The math in card counting is easy. Managing the process is incredibly difficult. So, why don't casinos just get rid of blackjack? Because it's incredibly popular. They also don't like to back off "advantage" or "skill" players because it looks bad. So, they've resorted to other means of drawing in the suckers and discouraging the card counters. Three important changes are (a) game variances, (b) changed payouts, and (c) automatic shuffling machines. The casinos are offering more "blackjack-like" table games. But these are horrible, with the odds stacked in favor of the house. The house has also implemented rule changes (like eliminating the surrender rule or what you can or cannot double-down on) and payouts. (The worst of these is the switch to 6:5 payouts for blackjacks. This reduces the card counter's edge, sometimes eliminating it and making the game not worthwhile to play.) Finally, some houses have introduced automatic shuffling machines to eliminate card counting. Period. What about all the other casino games, like craps, roulette, keno, the wheel of fortune, etc.? They're all bad bets with house odds stacked against you. Have fun. Gamble. But you're playing a losing game. Be comfortable with that.

You don't really need math to figure this out - just reason. If your game gives you a less than 50% chance of winning, you can still get "lucky" playing once. with every additional play, your chances of consistently maintaining your luck decreases, and quickly approaches zero. that's why casinos want you to stay.

I have worked as a dealer for 26 years. Never saw a big winner. If someone wins, most probably he goes again the next day, and looses all. My advice is STAY AWAY from gambling. It’s addictive. I have seen too many people breaking up and finishing with nothing.

My students often come up with questions how to win in casinos.... When I'm done with them, I have quite a number of shocked and disillusioned students that just learned how bad it is to be bad at math :-) PS: it I was carlos, I would buy the casino....

I am glad that you started uploading videos on math again

The amount of information you’ve providing in just this short, simple, explanation should not be understated. Thank you

It's always good knowing what percentage odds the specific casino will allow you to take on your pass line bet. And yeah, house blackjack rules can alter odds as well as the number of decks being used.

Great video. I find the 'Wizard of Odds' an excellent place to learn about gambling and the odds of going broke, which is more often than not when visiting a casino.

bruh i didn’t know you did this stuff i legit thought you only made skits on your other channel 😭 respect man

The grand daddy of expected value calculations for the card game Blackjack (21) can be found in the classic "Beat The Dealer" written by a mathematician who later made a fortune in hedge funds by finding pricing abnormalities. This 1966 book begat generations of card counters as well as famous rules like "always split aces and eights".

Thank you so much for such a clear and concise analysis of EV and the house edge. Even if the formulas make people’s eyes glaze over, once you point out the bottom line, it becomes so much more clear. When I go to a casino (pre-COVID days), I play poker. And depending on the day/time/etc., I may have to wait. And table games like these can be a decent way to pass the time as long as you know the odds and accept that you are going to lose. But the one thing you didn’t cover is: how slowly will you lose? You showed the EVs for the coin toss game, the pass line in craps, and red/black choice in roulette and how much one would lose over X hands (500, I believe), but you didn’t show how much time (on average) it would take to play that many hands. I think I saw something on one of your tables, but I would have liked a section towards the end of your presentation that said that, on average, you would play X number of hands per hour for each game and, therefore, lose X*Y/hour at each. And it’s that reason that, despite the math, I tend to play roulette while I wait. Red/black or Odd/Even might be 18/38, but the actual number of hands/hour is soooooooooo much slower than craps or blackjack that it’s a cheaper way for me to kill time. I have a decent grasp of basic blackjack strategy, but the game is much faster, and I’m usually trying to get my head into poker, so I don’t want to think too hard. To date, I’ve only tried craps a couple times and while I knew VERY basic strategy, the game went too fast for my taste. But if we are ever both at the same casino, I’d happily take some lessons and have some fun hanging with you and your brain!

Thank so much you, Zach!!! Subject of this video seems so paradoxical and kinda misleading. And that's probability and statistics is all about! It shakes your intuition and attempt to understand it hurts your brain! As always, a very informative and brilliant video!

Knowledge is indeed power and in my mind, this is, by far, one of the the best videos explaining what a player can do to attempt to come out ahead. Of course, luck will always be a factor! Some of these principles can apply to slots as well, however, one should be very careful when playing "progressives like slot games" e.g.,, the popular firelink and dancing drums. Regardless, any player must exercise "discipline" (as taught by the great Lombardi), try to play smarter (not harder), have a drive "to win", and finally, know when to eventually stop playing and quit (just like Brady!). Thank you for sharing this informative video!!

I have to adress a special thanks for the editor of this youtube channel, that was a revealing moment when I followed your interepretation the first time and that has consequences on my vision about risk and reward attribution. Absolutly MUST wathc for any mathematic passionate !

In my ECE stats class we went over a lot of gambling and writing MATLAB simulations of games deceiving expected values and things

1:20 "Just note: this is a probability problem so, yeah. It's a trap."

Good well delivered information. Long term proabilities. House edge. I found the charts helpful to understand the averages. Thank You!!

I feel my only power in a casino is being able to leave at any chosen time. If I get up in a session I can simply cash out and leave. Not many do though lol also the sessions where you never get up are the killers.

Thank you😊 for this video, do you offer account management or have any recommendations?

Our issue stems from the prevalent belief held by the majority that "having a good job is all it takes to become wealthy." Many people are unaware that these billionaires are using a different set of rules when conducting business.

Now I understand the relationship between time spent at the table or slot machine and comps - not necessarily just the coin in. Thank you for the info

More about house edge and the line bet on craps (from a career dealer, not a math head)... It is true that in any casino that offers table games, the best possible bets in terms of house edge are the line bets and come bets. However, reducing house edge *requires* betting the odds behind the line, and if one is serious about reducing house edge to the lowest possible degree, the player needs to bet the maximum odds allowed. The more odds the house allows (some only allow odds of 2x the pass line bet, others go as high as 100x), the lower the player can reduce the edge by betting the maximum odds. Come bets and line bets are essentially the same - a flat bet with odds for a number to repeat before a 7. The strategy decision the player has to make is how many numbers to cover. Three numbers is pretty standard, but your bankroll and risk tolerance are going to determine how extended you're willing to be on default.

A really excellent summary with easy to do math for figuring out the house edge. There is such a thing a beginner's luck. Bet once and walk away and you have a fair chance of winning. There is no such thing as long runner's luck. The more times you bet, in the long run, the more certain you are to lose.

My advice if you find yourself walking into a casino: don't

i remember making up a game way back, but i don't think i have the brains to calculate the expected return, it works like this: draw 9 cards from a deck with no jokers, and place them on a 3x3 grid if there is a line with three cards of the same suit, you lose if there is a line with three cards of the same COLOR (but not same suit), remove the card of different suit, and replace it. do this as many times as necessary until you either have three of the same suit in a row, or no more lines with three of the same color finally, if there are no more lines with three cards of the same color, you win. depending on the pattern the cards form, it's a different payout: house pays 3 to 1: R B R B R B B R B kite pays 2 to 1: R B R R B B B R R other pays 1 to 1: R B R R B B B R B

2:00 It was fun to sort out the logic behind this teaser in my head, thanks Zach!

"Never walk away from the table if you're on a heater". Good outro would be Bob Seger's _Still The Same._

wtf does he say at the very end? 'never walk away from the table if you dont have a heater?' - wtf?

A simple coin toss can show house edge. Natural odds for a coin toss are 50%, only allowing for proper outcomes. So, if a casino offers you $1.95 for heads on $1 bet and $0 for tails, using 10 flips as a proxy for infinity (note over 10 flips these will NOT be the results, but we're using 10 flips to proxy an infinite number of plays): Over these proxied 10 flips 5 will be H, 5T. Income is $10 (10 bets at $1 each), winnings paid will be 5 x 1.95 = 9.75. Thus the house keeps 25c of every $10 when the flips tend to infinity. This is a house edge of 2.5%. As the number of plays tends to infinity, the house is almost guaranteed to keep 2.5% of the bets taken. The casino has no control of the outcome of an individual bet, but they do have control over how much they OFFER you for a win, and this offer is always less than the natural odds. Slot machines don't 'get hot' or are about to pay out.

It seems to me generally that casinos tend to give games with higher variance payouts a higher house advantage. I don't know if anybody has quantified that relationship but there's some truth to it. So the greater the house advatage, the sweeter the game is to walk in with $100 and walk out with way more. Interestingly, as per initial argument. So myself? If i want to hit and run I'd choose roulette, and craps if im passing time.

I think you did a really good job of getting the important points across!

I learned a strategy a few years ago, but one that I think they'll ban you for. Every time you lose, double your bet. When you win, you'll make back at least what you've lost.

Simplest way to think about this, I think, is a smaller bet means you have to win more to reach the total before you walk away.

such a great vid!

I was a specialty contractor for many years. During the building of the casino, I was asked for a quote on a project nobody wanted and they didn't know how to do. I quoted them a cost plus price, plus % profit. They studied the proposal and everything was good. They found my profit percentage too high. So, I just said why don't we go with the same profit that the casino is projected to make while they are open and operating. They hund up the phone on me. I DIDN'T GET THE JOB.

“I know this is going to upset some people, but it’s true”. How true!!!

For those whon want to dive deeper into the mathematics: Until around 3:30 is the idea of stopping times After 9:00 is the theory of martingales :)

Great explanation. Spot on.

Aye been waiting for this one

I once heard a story of a guy who played the casino trying to walk away with $10. He would bet $10 and lose, then $20, etc. up to $510 or something, just to come out ahead by $10. Would this strategy work given you have a large money supply compared to the amount you are trying to earn?

There is an exception. Counting cards in blackjack, if you know how. It does take skill, but you can vary your bet based on what cards have been played and your hand and get up to a 2% advantage over the house. Now you'll probably get kicked out before you can take advantage of that but hey.

Interesting about the "Die" example. I used to be a roulette player (terrible house edge) but exciting then moved to Craps, which has 2 dice in play. As each die will average 3.5, added together that makes 7 - the House number (da-ding, lightbulb). Hence the reason you cannot consistently bet on 7 (compared to the other box numbers) and why lay bets (7 to win) require double the stake of back bets (betting on another box number) to gain the same return. I'm leaving out the Craps numbers, 11, Horn and other spread combinations for simplicity in the example. After 7, 6 and 8 have the next best chance of winning (and lowest return). End of the day you can mathematically strategy the game, but it comes down to blind luck or chance if you win. Although you can break even and make a little extra for tips if you use various methods. Not as much fun though! Hence the reason "The house always wins". I find the trick with casinos is to gamble for fun and casino points (for comps/rewards), rather than attempting to build a retirement fund. Doesn't take much to build up some free nights, free food, show tickets and any other freebies as the casino wants you to stay in the building as long as possible with hours of entertainment. The few that win are paid out from the majority that lose. Yes I've made the odd high (high for me) jackpot, which covered my trip costs. but I mostly "break even" or lose a little which is the cost of my entertainment. When it no longer becomes fun is when you have to quit!

U can rly change your life at the casino. Pull up in a 50k sports car, leave in a 150k bus

Thank you. With the help of this video, I’ve been banned from several national casino chains

Moved to Vegas when BJ had favorable rules. Left when those rules became difficult to find. Although to be fair, I am a poker player. Table games was just entertaining. But I knew every game. And the rules! And yes I even played Roulette for fun. Well kinda. I was involved in a wheel tracking system where a famous gambler (now barred from Vegas) tracked a wheel in the Gold Coast and the Paris. He found the wheel (must be hand made wood wheel) over years even decades of play developed anomalies. Depending on the big or small ball you could predict what numbers were more likely to land. Now I knew one of the team members and we attempted it at the TI but they switched wheels. The wood had a distinct pattern that appeared to look like an old woman in one corner. When we decided to pounce the wheel was switched. So the moral is I lost about 300.00 playing Tabasco machine on one minimum bet at a time. I tracked the roulette wheel results from Tabasco one spin at a time recording results. (This is allowed because everyone has a system in Vegas) (casinos love guys with systems) (not card counters though, another story) Then my partner was convinced the actual roulette wheel wasn’t switched (it was) we argued and basically flipped a coin. I lost the coin flip and the roulette wheel took 2k that night. Ahh good memories though. It was very fun to attempt to slay treasure Island even though my ass was handed to me. Poker grind and selling cars gets boring! Now I live in Nebraska and have a garden! The end!

@zachstar What do you mean by “never walk away from the table if you’re on the heater”?

Mathematician: The Casino always wins. The typical Gambler: No gamble no future.

Carlos is a good buddy of mine, we hang out on his yacht all the time.

Excellent basic math explanation of the house edge. I tell people that play roulette that they have better odds of throwing a twenty dollar bill out of the drivers side window & having it fly back in the rear window😂

We occasionally visit our local casino, never to win, purely for a fixed amount of entertainment, free buffet available, and free drinks, it's a good night out when controlled. A quick look around the establishment clearly shows who the only winner is.

The only game you can play in the casino with no house-edge is old-fashioned Poker (not the video/tablegames versions). And that's entirely because it's the one game where the house isn't actually playing (you only win or lose based on the other players at the table). The only money the casino makes from Poker is from their fees for hosting the table (rake/time)

Awesome video, man

The House has an edge which covers all kinds of expenses and overheads and (normally) even makes a profit. But we players have an advantage over them because we are (or should be) 100% in control as to when to play, how to play, how much or little we play, when to stop etc etc. In fact when it comes to systems/strategies they can never GUARANTEE you win, are never INFALLIBLE; but what they do is they make it easier for you to exercise your free will and have preset goals and limits so you don't get emotionally involved and take the wrong decisions.

want to be a millionaire? be a billionaire and enter the casino.

Hi zach ! Love this video ....I was looking for this video... I have a question. There was an old couple is united states who was able to crack lottery..have you heard about them ? .. I want to understand that math

Hey Zach awesome stuff here but where can I get a printout of the Casino house edge, thanks.

It depends on the money management strategy. Martingale, Fibanacci, etc changes those odds. Take a $1,000. Doing a Martingale, you will win $30 more times than if you would flat bet every spin/hand. Depends on the money management the player is using.

I have nothing against people going to a casino for a bit of fun. Take a few hundred dollars, play, most likely lose it all, and walk away. No real harm done. The problem comes when people think they can make money at a casino. STOP. You can't. Walk away.

My method for casinos is to treat them as arcades. Come in with a limited supply of funds. Anything you win in a game goes into your wallet separate from the money you brought in. Then at the end of the day if you brought x amount of dollars and expected to spend x but you got some back, then you got a bargain. Because if you change your view of it you change how you play it , and then enjoy it more or less.

Very informative. Thanks

Great video. But how do you calculate if you bet different amounts of money. Sometimes I’ll bet 10 dollar on blackjack but it can be that I double my next round

If you are good at card counting, you can get an edge in your favor of up to about 0.5% (in blackjack)

You have a gift! Your math matches almost exactly the English language statements you use. In other words, a person can see the math in your English, and see the English in your math.

great great video on casino gambling,, keep up

Your coin toss example was a perfect analogy for a slot machine, it is biased towards the negative (house edge) but under gaming law it has to get back to their programmed percentage return (98%, 85%, or however the machine is programmed) over a long period to make it a "fair" game. they are not quite as fair as a honest coin toss, but within the allowed gambling laws that apply This Return to Player (RTP) has many names but "balancing" is the best one I heard. The balancing period may be millions of spins, hence the reason some people lose over long term and others win on one pull of the handle. Slot machines are programmed to be more or less "loose" (volatility) by the casino (it is actually based on the percentage payback table and the denomination of the machine) to make it seem/sound like someone is constantly winning to encourage more players to take part. On average, slot machines account for roughly 70% of the casino's income, which if you factor in the electricity bill alone of the building is a staggering number. Low volatility machines pay back more often with smaller bets, high volatility pay occasionally with a large payback and the rest are somewhere in between. Only the casino and the manufacturer know the true dynamics of each machine and like real estate it is all about "location, location, location".

You can change the house edge to your favor by counting cards in blackjack, which is NOT cheating or illegal. But you should do your research before you try. You really have to get it perfect for it to work.

I thought of the Alice case via the simulated net profit lines such as at 10:10. It's gonna take her a long time to burn through that $10,000 so probably there will be a streak of rounds during which she has a gain of $100. Or probably not, when I saw that low probability for Alice's success I realized that getting 100 more wins than losses in some number of rounds might be much more rare than I thought. For comparison, you can see in the graph at 10:10 a gain of about $25 between rounds 500-600. The trick just is that it should happen pretty much right in the beginning, not when she's already lost some money. The graph I referenced was still losing money after that $25 gain.

I always assumed when I saw 97% payout that it meant you lost 3% everytime you bet - you'd lose everything over time

I was thinking of a roulette strategy that was to bet $10 on red and every time you lose bet three more times until red comes up again and start again with $10. The problem is that the amount of money you have to have so that a losing streak does not ruin you is very high.

These concepts are really valuable in financial markets as well. A lot of the trading community doesn't realize they're placing negative EV bets and they go bankrupt not even finding out!

Excellent video. Why do casinos continue to have crap tables if the house edge is much lower with this game? Also, takes maybe 4 or 5 employees to run the crap tables.

Do you know if manufacturers are required to publish the expected loss on all their slots? For instance can i find the house edge for Aristocrat slot games ?

Great video!!!

Best way to keep your money in a casino is to just keep walking right through.

Probability is one of the coolest maths

Excellent video : well explained

Well, you have totally convinced me to stay out of the casinos !

unfortunately i didnt get the exact last phrase. "Never walk away from the table....unless you're on the heater??" can someone explain? thanks.

Finally, a practical application for mathematics! Haha!

Brilliant video!!!!😍😍

I just realized this is the same guy as Zach star himself. I've been subbed to both for a while lmao

Of course. Statistically, you can't win at a casino. That's how the casino stays in business. My play: I start with $100. Then I DON'T go into the casino. I've won $100. Gambling is a tax on people who are bad at math. It's NOT fair. But people gamble for entertainment, not to win. They do it to momentarily have the illusion of winning. That's not an illusion that I want. But it's easy for me not to play - I hate the noise and congestion.

There is exactly one bet at the casino with no house edge. That is the "odds" bet on the craps table. The "odds" bet will pay you the exact odds on hitting that role making it a neutral EV bet. The casino takes a small edge on the 1 to 1 payout on the pass line itself (required in order to play the "odds" bet).

Of course, if you extrapolate any of those linear expectation graphs with negative slope, it will intersect the money=0 axis, and this is true for any non-zero house edge. In the house edge=0 case, it comes down to who has more money, you or the house (and guess what… it isn’t you).

Love it sooo much!! from Ethiopia❤❤

I love that line at the end where you basically just say "You should eat that paint if it tastes good" Some people you can't reach with reason and math, so you may as well just tell them to have fun 🤣

@4:10 My strategy is great. I play till I win 50, then take a break and play till I win 50 again 🤣🤣🤣

Man I had quit gambling for a while now. This vid makes my fingers itch...lol.

You can TECHNICALLY make the Come and Don’t Come bets and their respective odds at the craps table and enjoy the same EV as Pass/Don’t Pass, but I guess it’s the same game :-) fun and informative video!