Casino Looses 5%

[Freddy_the_Pig]

According to the story, the casino played about $600 million in bets, and the results deviated from expectation by about 5% of that, or $30 million.

We can get a feel for the likelihood of this by making some simplifying assumptions: (1) The only game the casino offers is a wager on a fair coin flip, with a fair payout; therefore, the casino’s expected profit is zero; and (2) Every gambler bets the same amount of money on each flip.

Let b = the dollar amount of each bet, and n = the number of bets in the quarter during which the casino lost money. Obviously, n = 600 million divided by b. We can then ask, for various values of b, what is the likelihood of a deviation in profit, from the expected value of 0, of at least $30 million?

It should be obvious that the likelihood rises as the bet size rises and the total number of bets falls. Mathematically, the standard deviation of the number of coin flips the casino wins is (n/4)^0.5. To achieve a deviation from expected profit of $30 million, we need a deviation in expected wins of 30 million divided by b.

If b = 1, then n = 600 million, and we need a result 2,449 standard deviations away from expectations. This is effectively an unobservable event. Likewise with any b up to 100,000, at which point a loss of $30 million is still 7.75 standard deviations away from normal.

Finally, with b = 1,000,000, we are 2.45 standard deviations away from normal, a result we will achieve (in one tail) 0.7% of the time. Still not common, but certainly possible. With b = 10,000,000, we are only 0.775 standard deviations away from normal, which will be achieved 21.9% of the time.

Conclusion: If the accounting and the games are honest, the casino must have been victimized by an unlikely, but not unimaginable, series of jackpots in the million-dollar range, or larger.

[B_A_Bay]

This is what people don’t understand, statistics and odds tell you what is LIKELY to happen, not what WILL HAPPEN. That is a difference

If I throw a penny 10 times, odds are it will come up heads 5 times and tails 5 times.

But there is nothing to say it won’t come up heads 10 times. It’s unlikely but possible. I could throw it a million times and it could come up head a million times. It’s not LIKELY but there is always that possiblity.

Slots are programmed to pay out, so that is not a game of chance. Black Jack requires some skill level, so that strictly isn’t a game of chance either.

In Las Vegas the slots everywhere are basically the same, because if one owner lowers the number of times his slots pay out, gamblers talk and will simply use his casino for slots. Of course if he lowers it too much he loses his shirt.

Gamblers have a well connected network and they know if a casino’s machine are too lose or too tight and visit those accordingly.

[Freddy_the_Pig]

@B_A_Bay

Slots are programmed to pay out, so that is not a game of chance.

That makes no sense whatsoever. Of course slots are a game of chance.

[Exapno_Mapcase]

@B_A_Bay

In Las Vegas the slots everywhere are basically the same, because if one owner lowers the number of times his slots pay out, gamblers talk and will simply use his casino for slots. Of course if he lowers it too much he loses his shirt.

This is wrong too.

Here’s the overall theory:

Generally speaking, loose machines can be defined as those that are programmed to pay back a greater percentage of the total monies played than similar machines elsewhere. But what is loose for one casino could be tight for another. For example, a loose machine in Casino A might be programmed to return 98% of money played, but in Casino B across the street a loose machine might be programmed to return just 92%. The term loose also has different meanings in different geographical locations. A return of 92% on £1 machine would be considered loose in London, but the same return would be considered tight in Reno. …

Many casinos advertise that they have “loose” machines with a 98% payback, but this statement, while narrowly true, can be misleading. Always examine that 98% manifesto with some suspicion. In small print, (make that microscopic lettering), is the expression “up to”, meaning that in the casino there are some machines, out of the thousands waiting there, that pay “up to” 98%.

And here are numbers specific to Vegas.

Most casinos won’t tell you how much their slots pay back, but some do. On the Vegas Strip, Stratosphere and Riviera have 98% dollar slots, and Circus Circus has 97.4% dollar slots. Fitzgeralds in Reno has a section of about 50 dollar slots whose average payback is 97.4%. If you’re playing dollar slots, I recommend that you play only at these casinos, or other casinos which advertise a specific high payback.

[x-ray_vision]

@B_A_Bay

This is what people don’t understand, statistics and odds tell you what is LIKELY to happen, not what WILL HAPPEN.

Which people?

[x-ray_vision]

@B_A_Bay

If I throw a penny 10 times, odds are it will come up heads 5 times and tails 5 times

Don’t mean to pile on, B_A_Bay, but this is incorrect also. If I remember correctly, the probability of that result is close to 1/4.

[muttrox]

@Shagnasty

We all know the story about the MIT guys and card counting.

In the book, they and a rival team descend on Mohegan Sun for their opening weekend and take it for millions. Management is new and doesn’t realize what has happened until they go over the numbers at the end of the first quarter. (Of course, the book seems to have played fast and loose with the truth, so who knows…)

[pulykamell]

@x-ray vision

Don’t mean to pile on, B_A_Bay, but this is incorrect also. If I remember correctly, the probability of that result is close to 1/4.

I suppose it depends what the definition of “odd are” is, but that’s still the most likely outcome, assuming a fair coin.

[x-ray_vision]

@pulykamell

I suppose it depends what the definition of “odd are” is,

Sure, but going by his phrasing, I’m betting he thinks (or thought) that result is most likely or has a 1/2 probability.

[counsel_wolf]

One point to note.

One should not consider the odds of this happening to just this particular casino but what the odds are that one casino in the US will lose in any given quarter, such a story would inevitably be picked up by the press. Once you enlarge the sample size in this way, it no longer seems like an unlikely occurrence, in fact I am surprised it is seen as unusual.

[pulykamell]

@x-ray vision

The eagle? What century you living in?

What’s the eagle? I’ve never seen that on any American roulette wheel.

[x-ray_vision]

@pulykamell

What’s the eagle? I’ve never seen that on any American roulette wheel.

In some forms of early American roulette wheels - as shown in the 1886 Hoyle gambling books, there were numbers 1 through 28, plus a single zero, a double zero, and an American Eagle.

Roulette

Other sources claim the eagle was used in place of the double zero.

[Jamaika_a_jamaikaiake]

@x-ray vision

Sure, but going by his phrasing, I’m betting he thinks (or thought) that result is most likely or has a 1/2 probability.

He was also right if he meant expected value. Just sayin’

Also, just to spread the knowledge, google has a calculator! I checked your 25% figure by googling:
(10 choose 5)/(2^10)
How cool is that?

[Pleonast]

@Sam Stone

It’s not just oversimplified, it’s completely wrong. You’re talking about ‘gambler’s ruin’, and the fact that a finite bankroll will crash to zero if you overbet it. Even winning gamblers like card counters and poker players face gambler’s ruin if they overbet their bankrolls.

However, while this affects the individual gambler, it does nothing to the casino. As far as the casino is concerned, there could be 10,000 gamblers in the place, each playing 1000 spins of roulette, or one gambler playing 10 million spins. It just doesn’t matter. The house’s profit will converge on 5.26% X the total amount of money bet, regardless of whether some places bust their bankrolls or not. In fact, ‘gambler’s ruin’ actually hurts the casino, in the sense that they’d rather see you mortgage your house and continue playing than lose your smaller bankroll and leave.

This is one of the more pernicious gambler’s fallacies (along with ‘locking in’ your winnings, and quitting while ahead to increase your profits). It sounds logical, but is completely irrelevant.

I think you are misunderstanding what Schnitte was saying.

Let’s say there are two gamblers, each with $100 in hand. They both play roulette, placing $10 bets on red. Gambler Alex plays until he has no money left. Gambler Betty plays until she has $200 in hand, or nothing. What’s the casino’s expected earnings on each gambler? Clearly, the casino will earn $100 from Alex, and expects something less than that from Betty, since she has a finite chance of reaching her goal.

You’re right that the expected earnings per roulette spin are constant, but the number of spins depends on when a gambler quits playing.

[TazMan]

@Pleonast

I think you are misunderstanding what Schnitte was saying.

I don’t think he misunderstood at all. Lets look at why Schnitte was wrong:

If gamblers continued playing indefinitely, the house would, on the long run, make a profit approaching the theoretical house edge (which, in double-zero roulette is 2/38 or about 5.26 % and in blackjack is between 0.5 and 1 %, depending on the rules and disregarding card counting). But that’s not the way gamblers behave, at least not the majority of them.

He’s claiming that casinos will make 5.26% on roulette only if they play indefinitely and a gambler’s behavior will change the house edge. This is wrong. They have an expectation of making 5.26% of all money gambled on roulette regardless of strategy.

Back to Alex and Betty.
If there were the same number of players employing each strategy and all Alexs went to one casino and all Bettys went to another with equal frequency, will one casino make more money than the other? Sure. But this would imply that Alexs keep going back with more money even though they lose every time and Bettys don’t decide to visit the casino more often as they have lost less per visit. An unlikely scenario. Casinos that have Alexs visiting also have to have more employees dedicated to more roulette wheels, serve more free drinks, etc., as there will be more spins of the wheel.

[RedSwinglineOne]

Here is a much more complete article,

Gamblers with seven-digit bankrolls took more away from Mohegan Sun table games than usual this spring, making for “an extremely long streak of bad luck” …

with better numbers.

Although players played $611.2 million at table games during the third quarter, up 6.4 percent, the casino kept about 11.6 percent, or 4.9 percent less, from players than it did last year. That drove revenue from table games down 25.4 percent from $100.9 million in third quarter 2007 to $75.3 million this year.

It only takes a whale, the term for the highest of the high rollers, to win at such a game as baccarat, where the casino’s advantage is slim, to offset the losses of other players making smaller bets, said Tamburin. That’s a chance casinos are willing to take, he said.

[Sam_Stone]

@Pleonast

I think you are misunderstanding what Schnitte was saying.

Let’s say there are two gamblers, each with $100 in hand. They both play roulette, placing $10 bets on red. Gambler Alex plays until he has no money left. Gambler Betty plays until she has $200 in hand, or nothing. What’s the casino’s expected earnings on each gambler? Clearly, the casino will earn $100 from Alex, and expects something less than that from Betty, since she has a finite chance of reaching her goal.

You’re right that the expected earnings per roulette spin are constant, but the number of spins depends on when a gambler quits playing.

No, what Schnitte was saying was that the house will make more than 5.26%, because players are more likely to quit while behind because they bust out than to quit while they are ahead.

I’ve heard this argument many times. The notion is that if you go into the casino with $10 and bet $10/hand in a game where the casino only has a 1% edge, the casino stands to make a lot more than that off of you, because if you play and lose, the casino makes 100% because you have to leave and have no chance of getting it back, but if you stay and keep playing, at some point you’re still likely to hit a bad run and bust out of the game. Therefore, the casino makes more than 1% when it attracts lots of players with small bankrolls.

It’s just not true. It’s the reverse of the logic that says you can improve your profitability by always ‘locking in’ your winnings by quitting while you’re ahead, or only playing with the ‘house’s money’ by putting your original bankroll away once you’re up any amount of money, etc. It’s sounds compelling, but it’s not. In fact, all you achieve by doing this is change the distribution of your wins and losses. If you always ‘lock in’ your winnings and quit while you’re ahead, the result will be greater than 50% of your sessions being winning sessions, but they will be smaller than they otherwise would be, and the losing sessions just as big or bigger as you chase that win.

In the casino’s case, the distribution of wins vs losses is the opposite for players with small bankrolls. The player will bust out much more than 50% of the time, but when the playser busts out all they lose is $10. But some smaller percentage of the time the player will go on a streak and win much more than $10. Either way, the result eventually converges on the house’s expectation.

Or look at it this way: the casino doesn’t ‘see’ your bankroll. It just sees a whole lot of bets at the roulette table. It doesn’t care whether they all came from one gambler or one thousand. It will earn 5.26% X the amount of the bet X the number of bets, and that’s it. In fact, busting out hurts the casino because it removes your action. The casino would absolutely love it if everyone who played there had an unlimited bankroll, because it would get more action from them.

[Darth_Nader]

@Tapioca Dextrin

I’ve heard that smart casinos love losing big money. Seriously. It’s great publicity.

Isn’t that what Podkayne thought was going on?

[Schnitte]

@Sam Stone

Or look at it this way: the casino doesn’t ‘see’ your bankroll. It just sees a whole lot of bets at the roulette table. It doesn’t care whether they all came from one gambler or one thousand. It will earn 5.26% X the amount of the bet X the number of bets, and that’s it. In fact, busting out hurts the casino because it removes your action. The casino would absolutely love it if everyone who played there had an unlimited bankroll, because it would get more action from them.

I don’t want to nag on this, but Pleonast’s last sentence has something plausible to it:

You’re right that the expected earnings per roulette spin are constant, but the number of spins depends on when a gambler quits playing.

The 5.26 % house edge which the casino will make on average is invariable, but the player can of course affect the number of spins he’s playing and such the total of his stakes.
A player with an initial capital of $100 who chooses a strategy of “locking in” previous winning and leaving as soon as he’s back at $100 will, on average, play much less hands than another player who keeps on playing until he has either $200 or nothing at all. The latter player will probably bet his initial $100 many times over, giving the casino 5.26 % of the total of stakes in all his games, whereas the former player will give the casino 5.26 % of his total stakes, which are, however, likely to be less than $100. In both cases, the house edge is 5.26 % of the total of stakes, but in proportion to the initial capital, the make-or-break player will be more profitable to the house.

[Sam_Stone]

@Schnitte

I don’t want to nag on this, but Pleonast’s last sentence has something plausible to it:

The 5.26 % house edge which the casino will make on average is invariable, but the player can of course affect the number of spins he’s playing and such the total of his stakes.
A player with an initial capital of $100 who chooses a strategy of “locking in” previous winning and leaving as soon as he’s back at $100 will, on average, play much less hands than another player who keeps on playing until he has either $200 or nothing at all. The latter player will probably bet his initial $100 many times over, giving the casino 5.26 % of the total of stakes in all his games, whereas the former player will give the casino 5.26 % of his total stakes, which are, however, likely to be less than $100. In both cases, the house edge is 5.26 % of the total of stakes, but in proportion to the initial capital, the make-or-break player will be more profitable to the house.

If all you’re saying is that the longer you play, the more money the house will make, we have no argument.

The house has an expectation of winning 5.26% X the number of bets you make X the average amount of money on each bet. If you place $10,000 in bets, they’ll make more than if you place $1,000 in bets. No question about it.

Players who ‘lock in’ their winnings may lose less than players who keep playing, but only because they don’t bet as much money. They’d do even better if they just stayed home and didn’t play at all. But the idea of gambling is either to win or to entertain yourself. If you play Roulette you can’t win, so you’re playing for entertainment. That being the case, rather than betting $10/spin and ‘locking in’ your winnings, you’d be better off playing $1/spin and playing 10 times as long. It costs you the same, and you get ten times the entertainment.

Usually, when people tell me about extra money the casino makes because players bust out, they believe that the casino actually gets more than the 5.26% house expectation on the bets. They don’t.

But clearly, if you keep recycling your money until it’s all gone, you’ll be broke. But the casino didn’t make more than 5.26% - they made the percentage, but you just put a lot more money into play by recycling your bankroll until it was gone. But recycling your bankroll 20 times until it diminishes away looks no different to the house than having 20 players each play through their bankroll once and leave with their winnings or losses.