Albert Hibbs
(With Some Exposition on Casino Odds
and - ah - "Activities")

On occasion the author and illustrator of CooperToons has been asked to assist his colleagues at various tasks. The assistance could be as simple as devising a computer algorithm for choosing a restaurant suitable for some fest. Other help might be to provide numerical analysis for interpretation of experimental data. Such requests were never refused.

Weeeeeeellllllllll, like the Captain of the Pinafore, we have to qualify that statement. Once there was a request to calculate the sequence of numbers that would come up on a roulette wheel. That way, the supplicant smiled, he would be able to go to a casino and beat the wheel.

Stifling the impulse to bray with uncouth laughter, the same-said author and illustrator of CooperToons pointed out that the prediction musts needs be probabilistic. That is, even if you could guess the correct number better than chance, the prediction would also have to be good enough to overcome the house percentage.

"What," the petitioner asked, "is the house percentage?"

Such is music to casino owners' collective ears. Nothing warms their hearts - or their pocketbooks - more than - ah - "guests" who are fledglings to the principles of casino gambling.

OK. So what are those principles? We'd really like to know what they are.

I thought you would as Captain Mephisto said to Sydney Brand. It's very simple really.

Go to a friend and hand him $10. Then have him hand you $9 back.

Then give him your $9. Have him give you $8 back.

Keep this up until he has all your money.

That, ladies and gentlemen, is casino gambling. Whenever the dealer should pay you a certain amount, he keeps a portion for himself. That portion is called the house percentage.

The house percentage varies with the game, of course, and often with the type of bet. For instance, in roulette there are 18 red numbers, 18 black numbers, and on the traditional American wheel, 2 green numbers (zero and double zero).

Now such a design is craftily qualified to give the house better payoffs than the player. Suppose you bet $5 on red. If you win, the house will give you $5. So that's getting paid at 1:1 or even money.

However, the true odds of getting red are not 1:1. That's because there are 18 red numbers, yes, but there are 2 green and 18 black numbers. So there are 2 + 18 = 20 numbers that are not red. So the true odds of getting red are not 1:1 but:

True Odds (Red)		=		Total Red : (Total Black + Total Green)
		=		18 : 20
		=		1 : 1.11

For $5 that's:

True Odds (Red, $5) = 5 : 5.55

So if you bet $5 and won, the casino should have forked over another 55¢.

The house percentage, though, is usually - as the name implies - expressed as a percentage.

The percentage, though, is not simply the 55¢ divided by the $5 bet. That would be 0.55/5 = 0.11 or 11%. No, instead you also have to consider both how much you should win and with how much you lose.

With our odds of 18:20 to win, then that means on the average you will win 18 times out of a total of 38 spins. So the probability that the player wins betting red (or black) is:

P(Player)		=		18 ÷ 38
		=		0.47368

... or about 47 %

The probability that the house would win is then 20 out of 38 spins or:

P(House)		=		20 ÷ 38
		=		0.52632

... nearly 53 %.

To summarize if the player bets $5 on red, the house will have to fork over $5 when the player wins. But that happens only 47 % of the time. And the house will pull in its $5 about 53% of the time.

So how much dough will the house win on the average? Well, it's the amount they take in minus the amount they pay out.

House Win ($5 Bet)		=		P(House) × Player's Bet - P(Player) × Player Payoff
		=		0.52632 × ($5) - 0.47368 × $5
		=		0.26316

... or 26¢ on the $5 bet.

So we calculate the house percentage as:

House Percentage		=		0.26316 ÷ $5
		=		0.05263

... or 5.26 %.

OK. But what about some other bets? What about a straight number bet?

We can calculate this percentage exactly as we did before. But the probabilities and the payouts differ.

The payoff on a straight number win is 35:1. So if you bet $5 and win, you get paid at 35 × 5 or $175.

But there are still 38 numbers: 1 through 36 and zero and double zero (we're still talking about a traditional American wheel). So the odds of picking a single number is:

P(Player)		=		1 ÷ 38
		=		0.026316

... or 2.6 %

On the average, then, the house will win 37 out of 38 spins. This means the probability the player loses is:

P(House)		=		37 ÷ 38
		=		0.96368

... or 96 %

So with a $5 bet:

House Win (Straight Number, $5 Bet)		=		P(House) × Player's Bet - P(Player) × Player Payoff
		=		0.96368 × ($5) - 0.026316 × $175
		=		4.86842 - 4.60526
		=		0.26316

So the percentage is:

House Percentage		=		0.26316 ÷ $5
		=		0.05263

... or 5.26 %

Exactly the same as betting on red or black.

In fact, with one exception every bet in roulette - single numbers, zero, double zero, red, black, odd, even, high, low, columns, and rows, all have this 5.26% house percentage. That sole exception is the five-way combination. This is where you bet that the next spin will land on zero, double-zero, one, two, or three. Each time you bet $5 and win, the dealer hands you $30. Therefore the bet pays 6:1. But if you were paid at true odds he'd give you $33.

This payoff produces a house percentage of 7.89%. The proof of this will be left, as the textbooks say, as an exercise for the reader.

Because the payoffs in casino games are tilted in the house's favor, the house percentage is often called the house advantage. Other terms are the house edge, the house take, the cut, the vigorish, or simply the vig. The last word may be useful when you're playing Scrabble.

For a lot of players, having a few percent against you doesn't seem too bad for an evening's entertainment. What if we play for a few hours and may lose a few bucks? And indeed you may read, not only on famous informational websites but even in some columns by bonafide experts that the house percentage is the average amount the house wins and is the average amount the player can expect to lose. Strictly speaking this is not correct.

The house percentage is not the player's average loss, but the player's average rate of loss. That is, if you keep playing, then your money will eventually get whittled down until you lose it all.

But wait a minute, you say. Isn't the house percentage something that is important only in the long run? Suppose you play only a few hands. Then there's not enough bets for the house advantage to have an effect. After all, if you bet once on a straight number and win, the house loses and gets none of your dough. Where's their percentage?

Soddy, old chaps. Remember, if you win on a straight number, you get paid $35 for a $1 bet. But you should have been paid $37. The casino always has their percentage even when you win.

There is, though, one wee little thing that could tip the house percentage. Suppose, just suppose, you could predict where the ball lands. Maybe not perfectly, but suppose that instead of guessing the number an average of 1 out of 38 times, you could predict a number 1 out of 30 times. What would the house percentage be then?

Remember you're still playing by the usual rules. So when you win, you would still get paid 35:1. But that would not be 1 out of 38 spins, but 1 out of 30. And of course you'd lose 29 out of 30 spins.

.

So let's say you bet our $5 on the number. So using the new odds and the old payouts we get:

House Percentage		=		P(House) × Bet - P(Player) × Payout
		=		+ (29/30) × 5 - (1/30) × $175
		=		4.8833 - 5.8833
		=		-$1

Note - and pardon us if we shout:

THE HOUSE PERCENTAGE IS NEGATIVE!!!!!!!

Specifically that means, the house loses $1 on a $5 bet.

So we should be talking now about the player's percentage:

Player's Percentage		=		1 ÷ $5
		=		0.2

... and you, the player now has a favorable percentage of 20%.

So if you could guess a number correctly 1 out of 30 times, then you would overcome the house percentage and have a net positive expectation. And a pretty substantial one at that.

Which in 1947 is what Albert Hibbs and his friend, Roy Walford, said they did. While they were students at the University of Chicago (Albert in mathematics, Roy in medicine), they went to Reno. They said they could predict that a number would show up with a greater frequency than chance. So they played the number.

So what happened? Or specifically, how much did they start out with, and how much did they win?

Well, like all indisputable facts and incontestable tales, the information is confusing and contradictory and the truth depends on who's doing the telling. In some stories you read Albert and Roy started out with $100. In other tellings it was $200 or perhaps $300.

As to how much they won, well, that varies too. Some stories say it was $4000. Or maybe it was $5000. Or $6000. $8,000? $14,000? Wait a minute! How about $30,000? No, wasn't it $40,000?

When confronted with such discrepancies historians have to turn to primary source material. Or if you can't find a true primary source - information that came from the time of the event and from someone who was there - you have to settle for stories that are the earliest and where the teller is closer to the actual participants. Then you should get at least some semblance of the truth.

We wish it was so easy. Although there are books that give detailed accounts of Albert and Roy's trips to Nevada, it's not easy to track down the actual sources. Worse, the lack of definitive sources leads to careless errors which cause confustication.

For instance, one of today's biggest and most important financial magazines recently wrote about Albert and Roy. Yet they cite a source which strictly speaking doesn't exist.

Now if you want to be charitable, you can say that they simply cited a source, but they missed the date by two years. You may think this is quibbling. So the writers (or editors) made a mistake. So what?

Well, look at it this way. If a magazine can't even get a simple date correct - a date of an article that even a middle school student could find by the simplest of Internet searches - how much confidence can we have regarding other information in the article - particularly information that arises out of complex sifting through even more obscure sources?

Perhaps the most accessible primary source is Albert's personal, albeit brief, account. He told his story on January 22, 1959.

On that day Albert was a contestant on You Bet Your Life, the quiz program hosted by Groucho Marx (the secret word, by the way, was "voice"). Albert said he and Roy cleared about $12,000. They then used the money to buy a yacht and sail around the Caribbean.

Although Albert's answer was given twelve years after the event, you'd still think his answer would settle the question. And there is general agreement (with some qualifications) with one of the earliest and most accessible printed accounts. This is a brief - very brief - article in Life Magazine from 1947. The story said that at first Albert and Roy won $5,000 in 40 hours. The casino manager got a bit antsy and so switched the wheel. Then they went to another casino and their money rose to $14,500.

At that point we read that "unaccountably" the system "went sour". They began to loose and they quit when they still had $6,500.

Perhaps, then, when Albert said they made $12,000 he was referring to the peak amount they took in. Sure, he may have been off by a couple of grand, but he was speaking more than a decade after the fact (and also sparring verbally with Groucho). And he did say they made "about" $12,000.

But before you rush out and try to find a biased wheel, there's a number of points to consider. At the time Albert and Roy were in Reno and Vegas, a number of seasoned gamblers saw nothing unusual in their play. After all, Albert and Roy had played roulette. They won for a while, then lost, and left before they went broke.

Albert and Roy's wins, the gamblers said, was simply luck. A manager of one of the biggest casinos on the Las Vegas Strip remembered seeing a player pull $20 out of his pocket and end up winning over $20,000 in a few hours. And after Albert and Roy had gone back home, one casino owner even smilingly invited them to come in and try to win at his casino.

Adding to the confusion is that some sources also tell us that there was a second trip in 1948 where Albert and Roy returned to Nevada and tried their system again. They were doing so rotten that they were about to give up in despair when the owners of another casino asked them to come over. And they, the owners, would bankroll them.

Ha? (To quote Shakespeare.) Why would the owners do that?

It was simply that they believed the publicity garnered for the casino would bring in more money than what Albert and Roy would win. If they won, then the news stories would draw more players than ever. And if Albert and Roy lost, well, the owner's money would just go back to the casino.

The story as sometimes told is that at the new casino Albert and Roy began to win. They won so much that the bosses decided to call the deal off. But they did let Albert and Roy keep their winnings. Albert and Roy, now getting quite weary, readily agreed. In the end, the two men left with a good, if not extraordinary, profit. And they bought their boat.

OK, just what exactly did Albert and Roy do?

Now today you read how a lot of people "beat the wheel" by using small hidden computers that can time the speed of the ball and the wheel. The computer then predicts a number that the ball will end up on. Since you can lay bets after the ball starts spinning, the computer calculates the number and the player then bets on the predicted number. As long as the prediction is accurate enough to overcome the house percentage, the player should come out ahead.

It's hard to tell who was the first to try this scheme or how much they won (or lost) as a number of people wrote that they tried it out. Some were (and are) established (even iconic) mathematicians at major universities and research facilities. The time frame also varies from around 1960 into the 1970's.

As you may expect the our-computer-beat-the-casino stories can be fraught with inconsistencies and contradictions. In some accounts you'll read the computer experts walked off with thousands or even millions of dollars. But other stories tell more of the technical problems encountered and why the would-be casino-beaters had to abandon the project without winning really big money.

Albert and Roy, though, did not use a computer. Instead they said they found a biased wheel. That is, they identified a wheel that did not produce a truly random sequence of numbers.

But what would cause a non-random wheel? After all, the bouncing ball could land on any number, and it would then bounce up and rattle around before coming to rest. That's pretty random.

Well, a lack of randomness could be due to physical imperfections like cracked surfaces, dented deflectors, worn slots or dividers (called "frets"), or the wheel not being completely level. Also roulette wheels are expensive and can't be replaced several times a shift like a pair of dice or a deck of cards. A wheel might be used by a casino for years.

The way you identify a biased wheel, Albert said, was to watch the play and write down the numbers that come up. You could then determine which numbers appeared most often. True, you have to do this long enough - on You Bet Your Life, Albert said it took months - to be sure you are seeing a true systematic error and not just a chance clustering of certain numbers over the short run.

Now attempts at "clocking" roulette wheels are almost as old as roulette wheels themselves. Or at least the practice goes back to 1873. In that year we read that an Englishman named Jaggers (his first name varies with the telling) hired a bunch of clerks to watch the roulette wheels in Monte Carlo. Their clocking revealed some numbers which showed up more than they should. By betting on these numbers we read that Jaggers cleaned up. Of course, the amount won varies with the telling but it's usually claimed to be at least 2 million francs.

Some people, though, pooh-pooh wheel clocking or other methods claimed by "advantage" players. They say that although in the real world randomness isn't perfect, it's random enough. Look at it this way. From the way some clockers talk you would think casinos are filled with lopsided wheels with rattling twisted frets where the ball careens around gouged and scratched surfaces until it plonks down on only a few numbers. And through all this the casino owners don't notice a thing.

So what actually went on during Albert and Roy gambling sojourns?

We can't dispute that Albert and Roy may indeed have found a biased wheel that helped them win. After all, when Albert and Roy played in the late 1940's, casinos did not have the rigid quality control and maintenance safeguards they have today.

But on the other hand, the fact that they also had bad luck - at times very bad luck - raises the possibility that maybe they were just lucky. And being smart - Albert and Roy quit while ahead.

But wait a minute, you say. The way you talk about the house percentage, no players should win. But what about my brother-in-law, Bubba? He goes to the Las Vegas with his wife once a year. For the last five years, they've spent a long weekend there and he says they play a system. They have a bankroll of $500 and always stop when they make $100. They've always come out ahead. And you won't question my brother-in-law Bubba, will you?

Not necessarily. But with games of chance you see oddball things that make the game look better than it really is. For instance, if you have a large bankroll and bet low amounts on even-money bets, then you certainly might see a gain before you lose it all. You might be lucky and even show a profit over some time.

Or to put it another way. Suppose we have thousands of Bubbas who go to Las Vegas for a weekend. They vow to quit gambling when they win $100. They all see some sights, catch a couple of shows, visit the Hoover Dam, and spend a few evenings playing the games. For the next five years, the Bubbas return to Vegas for their weekend.

That at least some of the Bubbas show a net gain is certainly possible even with the house advantage working against them. And it's those Bubbas who tell you how they have consistently won. Losing Bubba's - who are the majority - rarely brag. But the casino still gets most of the moolah.

Now there's no doubt that Albert and Roy were playing an honest game, but there's one thing all prospective gamblers should ponder when hearing about a marvelous amazing new system that really, really works.

In some cases consistent winning might mean there is simply skullduggery going on. Today a few hundred casino employees are arrested every year for cheating. They range from the lowest level employees to the managers (in fact, over 20% of the arrests are at the managerial level). Some of the cheating rings have taken casinos for millions of dollars.

Of course, to waylay suspicions that you're cheating you just claim you have a new system. And unless you catch the cheaters red-handed it's hard to disprove the "system" argument. One famous gambling expert completely pooh-poohed a claim that a big winner had a new system to - quote - "break the bank" - unquote - at a hole-card game. The Famous Expert said that he had interviewed the men who had put up the money (and who had also been the actual players). He said they privately admitted they ignored any system and had simply bribed the dealer to signal his hole card.

But in roulette? Can you actually cheat?

Yes, and it's also easiest with a little help from the croupier. Top-hatting is a term used for several methods of cheating. The simplest topping is where the dealer simply adds extra chips to the stack of a player when the player wins. Say a player has five chips on red and wins. So the dealer slips him 7 chips instead of five. So it turns out the player now has an advantage of:

Player's Win		=		7×(18/38) + (-5)×(20/38)
		=		0.6842

... and as a percentage:

Player Percentage		=		0.6842 ÷ $5
		=		0.14

... or 14 %! So the player cleans up and after the game splits his winnings with the dealers.

Another cheating technique - again with a friendly croupier - is for the dealer to put a bit of adhesive on his finger. Then he'll put a little dab on the slot of a number. Now a little dab would do you - the goal is to have the ball hit that slot and the neighboring numbers more often than chance. But if the slot was so sticky that the ball sticks only on one number spin after spin, then even the most trusting pit boss would think something was amiss.

Needless to say, any form of cheating is not only very inadvisable, but it's just plain stupid. Today casino tables are continuously monitored with closed circuit TV and the consequences can be serious. In the - quote - "good old days" - when the mob openly controlled the casinos - not as long ago as you may think - the owners didn't bother with the pesky trifles like calling in the cops. They'd break your fingers, smash your kneecaps, or just pummel you into insensibility. And that's if you only stole a few bucks. If you got caught snitching the big money, well, you could go fishing with Luca Brasi.

OK. Enough already! Just who was Albert Hibbs?

Albert Roach Hibbs was born in Akron, Ohio in 1924. His dad was an engineer and his mom's education and profession showed that she was a woman of extraordinary ability, intelligence, and talent. With such parentage, it's no surprise that Albert became interested in science (and science fiction) as a kid. He later attended CalTech and graduated with a bachelors degree in physics in only three years. He then attended the University of Chicago were he picked up a masters degree in mathematics. He returned to CalTech where he got a Ph. D. in physics under the direction of Richard Feynman who would later win the Nobel Prize in Physics.

After graduating, Albert went to the Jet Propulsion Laboratory. He originally worked on guided missiles - a big field in the early Cold War - and later became responsible for the orbital calculations for the satellite program. And when the JPL fell under the umbrella of NASA, Albert began working on the planetary missions.

Those who remember the early days of the space program can't forget that failed missions - crashing rockets, exploding launch pads, and such stuff - were common. Fortunately, this was the day before difficulty in pioneering research would be seized by politicians for their own self-aggrandizement and the space program continued. Albert remained at the Jet Propulsion Laboratory where he retired as director in 1986.

With his wry personality, Albert was a natural for talking with reporters and he would appear on television shows, including Exploring. This was a science program for kids which is a programming genre that has almost vanished (contrary to increasingly popular belief, shows about ancient aliens, non-existent sea monsters, and mermaids are not science programs). Albert shared a similar sense of humor to his advisor, and he wrote the introduction to the well-known anecdotal biography of Richard, Surely You're Joking, Mr. Feynman.

Given Albert's public prominence as part of NASA programs, it's a bit ironic that today his most famous film clip is the re-run of the You Bet Your Life show. As usual Groucho tried to throw the contestants off with his wisecracks but Albert's ability for performance and his quick mind stood him in good stead. When asked where he was from, Albert said he was from Ohio, around Arkon and Chillicothe.

"Chillicothe?" Groucho said. "Isn't that half lima beans and half corn?"

"No," said Albert, "It's halfway between Columbus and the Ohio River."

When asked what he was doing, Albert said he was a physicist at the Jet Propulsion laboratory. Groucho asked what else he did. "Well, for the last few years," Albert added, "I've been working on writing a book. It's called Quantum Mechanics and Path Integrals."

Groucho asked what the plot was about.

"Well, the plot", Albert said, "is a mathematical exposition of quantum mechanics using a path integral approach and probability amplitudes. This also has applications in other fields."

After a pause, Albert asked:

"Put you down for a copy?"

Groucho pondered a moment and said, "Why don't you send me a copy and I'll give it a plug on the Pinky Lee Show."

The topic Albert and his fellow contestant, Mamie Masara, picked was "Earth, Sea, and Sky". At one point Groucho asked "What is the popular name for the constellation 'Ursa major'."

"The Big Dipper", Albert replied.

"It's also called the 'Big Bear'", Groucho said.

"That's when she goes down to bed at night and takes her clothes off", Groucho added (Groucho loved puns)."

"I thought she came up at night," Albert commented.

"You know," Groucho said, "if you want my job, you can have it."

The good news was that Albert and Mamie answered all their questions and won $1000. Then they decided to go for the bonus question. This was a wager for half their winnings and could net them $5,000.

The question was what was the name of the Queen of Greece who had just visited Washington. Unfortunately, this was one gamble that didn't pay off. Albert and Mamie's answer, "Helen", was wrong as the queen's name was Frederica. Albert and Mamie did, at least, walk out with $500. So like in Vegas and Reno in 1947, at least Albert came out ahead.

References

Departing from the usual policy of explicit referencing source material, this essay relies on a number of implicit references regarding casino gambling. Part of this choice was because many of the books pretty much give you the same information that you can find elsewhere. Also by a lack of explicit citations the author and illustrator of CooperToons hopes to avoid any appearance of endorsing any particular claims of a given author on how to win (or lose) at gambling games.

"Albert Roach Hibbs", William H. Pickering, Physics Today Volume 57, Issue 1, p. 68, 2004

"How to Win $6,500", Life Magazine, December 8, 1947. The article with the correct date.

"Boys Quit Reno While Ahead $8000", St. Petersburg Times Magazine, November 23, 1947.

"Albert Hibbs, 78; JPL Scientist, Voice of Unmanned Missions", Myrna Oliver, The Los Angeles Times, February 27, 2003.

"Albert Hibbs, Mathematician", Times of London, February 27, 2003.

You Bet Your Life, Groucho Marx (Host), Albert Hibbs (Contestant), National Broadcasting Corporation, January 22, 1959, Internet Movie Data Base.

Understanding Probability: Chance Rules in Everyday Life, Henk Tijms, Cambridge University Press, 2004, (Third Edition, 2012).

Practical Statistics for Astronomers, J. V. Wall and C. R. Jenkins, Cambridge University Press, 2012.

Scarne's New Complete Guide to Gambling, John Scarne, Simon and Schuster, First Edition (1961), Second Edition (1974). The first and second editions have major differences, some of which are controversial. This reference is included as there are no claims of winning systems and John points out correctly that any system betting with a negative expectation must lose in the long run. That said, more recent research has found that some of John's later claims - such as showing Bugsy Siegel how to count cards in the 1940's - can be disputed as they contradict some of his other writings and claims.

The Big Book of Blackjack, Arnold Snyder, Cardoza, 2006. This book has a discussion of the controversy of John Scarne vs. other gambling advisors such as Edward Thorp, Allan Wilson, and Michael MacDougall.

The Weekend Gambler's Handbook,, Major Riddle and Joe Hyams, Random House, New York, 1963. Although this is an interesting and amusing book, we give it simply as an example of an early "how-to" book on gambling. However, the inclusion here is not an endorsement of the advice. For instance, Major (his real name, not a military rank) says you should "test your luck" when starting out by making a number of small bets. He also has a betting strategy which says you should bet more heavily when winning and lower your bets when losing. Both bits of advice are not only vague, but are based on the maturity of chances - that is the "gambler's fallacy".

"Cheater's Justice: The Brutal Payback for Duping Dealers at Casinos", Michael Kaplan, Maxim, October 15, 2015.

"Basic Casino Cheating Scams Hardest to Catch, Gaming Experts Say", Nicole Raz, Las Vegas, Review-Journal, October 15, 2015.