Note: this is a sketch of a talk I gave yesterday at a conference filled with computer programmers, network and software engineers.
This is Las Vegas, the land of magic. Penn & Teller, Siegfried & Roy, David Cooperfield. And now me. I won’t do any tricks, but I’ll let you in on a secret.
The way to make a woman disappear is to walk up to her and say, “Hi, I’m a statistician!” Poof! She vanishes!
Bad news is, unlike the tricks on the strip, there isn’t any way to make her undisappear. So use this trick sparingly.
Vegas is also the land of gambling, and gambling means probability.
Now you might figure that facility with figuring the odds—knowing the probabilities—is all you need to be a good gambler. This isn’t so. To make a living at gambling—and many do—you do need to figure the odds, yes, but you need to beat more than just the odds.
If you can’t consistently beat more than just the odds, you will go hungry. Then you’ll be forced to take up some less glamorous profession for your daily bread, like say programming a computer.
And if you thought saying, “I’m a statistician” worked magic, just wait until you’re forced to admit something like, “I manage a TCP/IP stack with a specialty in abstraction layers.” Walk down the streets of San Jose after Eight PM and you’ll see how efficient a trick this is.
So let’s figure how to figure the odds, and then figure more than the odds.
I was walking up First Avenue with a friend, minding my own business, when a barrage of crayons rained down on me. Crayons—as in Crayola.
I looked up in time to see a school bus roar past, the windows open, the kids laughing like maniacs.
My first thought was to the utility of restoring corporal punishment in schools. But my second was to probability, because my friend noticed that one of the crayons stuck in the crease of my hat. I was wearing a fedora, as all gentleman do. I believe it was an orange crayon, obviously bitten in half. My friend, pointing to the stuck crayon, and thinking himself hilarious, said, “What are the chances of that!”
Now, statisticians are always hearing that joke. And we’re always expected to smile at it, to pay a compliment to the comedian for his originality. But the feeling that joke induces is the same as the one meteorologists experience when they hear some wag say, for the eighteen-hundredth time, “Everybody talks about the weather, but nobody does anything about it!”
It is in these frequent moments we wished we did manage a TCP/IP stack, because no one has any idea what that is.
There are, though, plenty of inside TCP/IP jokes. Like, “An IPv6 packet walks into a bar. Nobody talks to him.”
If you understand that, we’re back to our magic trick.
Anyway, the answer to these what-are-the-chance questions never changes. It is always the same number. It is a number you all know very well. I’ll let you prove that to yourself in the few minutes.
Even stranger, though, is that that number is consistent with this truth: there is no such thing as probability (and therefore no such thing as odds). It doesn’t exist. It doesn’t have being. Probability is always a state of mind.
Probability is something you bring to a problem; it is not something already there, to be discovered. Neither is it subjective, something you make up. It’s fixed by your assumption.
Here is a simple demonstration of this paradoxical truth. Before I came up here, I took my hotel key card and put it into either my left or right jacket pocket. What’s the probability it’s in the left?
Those who said one-half were starting with these assumptions: the key card must be in one of two pockets, and the left is one of these pockets. The probability of left given those assumptions we deduce (correctly) as one-half.
But think. What if you saw me getting dressed? I saw me getting dressed, so such a horrific spectacle is not impossible. Then you’d have different assumptions. You’d know, like I know, which pocket the card is in. On your new assumptions, the probability is either 0 or 1, depending on what you saw.
So here are two separate probabilities for the same event. Both are correct, and both are products of what you bring to the situation.
As an aside, the failure to account for the differences in states of knowledge is responsible for the confusion over the infamous Monty Hall problem. Heard of that? Game show host says there’s a prize behind one of three doors. Contestant picks a door, say 1, and Monty opens another, say 2, and asks contestant if he’d like to switch to 3. What’s the probability, given these assumptions and none other, the prize is behind 3? Work on that in the back of your mind.
How about a coin flip? Probability of one-half for heads, right? No.
Only on the assumption, “this is a two-sided object which when flipped must show either heads or tails” is the probability one-half. On the assumption a physicist, who measures the spin and upward force, would make, the probability is (again) 0 or 1, depending on those measures. It turns out to be easy to measure the state of a coin.
Something (actually many things) causes that coin to come up heads, and those causes are related to spin and force. If we knew the spin and force, we’d be able to guess the outcome, and thus come to a different probability. This proves there are only unknown causes, and that probability, randomness, and chance do not exist. Randomness didn’t make the coin heads, physics did.
The lesson for both the key card and coin—for they are identical situations probabilistically—is not just that probability is not unique, but that the more you know, the better you are at guessing the outcome.
We’re getting closer to why you have to beat more than just the odds.
Let’s stay with the coin. The odds of heads are one to one on the standard assumption. Suppose we make a bet that I pay you a dollar if heads, or you pay me if tails. That’s fair, as long as neither of us knows or can control the causes of the flip—which is a dicey assumption. Some magicians are good at controlling these causes. That’s what magic is all about.
But what if I pay you only eighty cents if heads, and you still must pay me a dollar if tails? That fair? No? Why?
That mismatch in payment is the basis of all casino operations.
That mismatch is also why you have to beat more than just the odds to win. The odds of a head, given the simple assumption we’ve been working with, are 1 to 1. But with the mismatch in payment, it is like the probability has been adjusted to 44% instead of 50% for heads (or odds of 1.25 to 1 against). Heads will still come up roughly half the time, but since you’ll pay more on each loss, it is like heads only come up 44% of the time.
To win at coin flips, then, requires acquiring more knowledge about each flip, like a physicist might be able to do. You have to gather enough new information to not only make up that 6% deficit, but to exceed it, else the game will be roughly a tie between you and your opponent.
Barring specialized physics equipment, or lacking the talent of manipulating flips like magicians can, you won’t be able to do this. So you’ll have to find a new game.
How about Blackjack? You have to make total of 21 or beat the dealer without “busting”. To make this easy, suppose there is only one deck. You’re the only player and have just been dealt a Jack, and the dealer shows an 8. What is the probability you get a blackjack with your second card? Given the usual assumptions, it is 4/50, or odds of 11.5 to 1 against, because there are 4 Aces in the 50 cards left and you need one of them.
But what if you got a peek at the bottom card of the deck after the dealer shuffled it and was putting it into the shoe? If that card was an Ace, then your chance is lower, because you can’t possibly be dealt that Ace. But if were any other card, your chance is higher, because you’ve effectively removed that card from the deck.
Instead of 4/50, the probability is 4/49: 8% to 8.2%. Seems like a small difference, but that kind of edge works in your favor over many hands.
This is also why card counting works. Card-counts acquire that extra information you need to beat more than the odds.
But you have to be really good at counting, because the edge it gives you even if you play perfectly, is only around 1%. It’s worse than that, because casinos are onto the usual tricks of card counters. Many are now using continuous shuffling machines to reshuffle the used cards after each hand, which removes most of the advantages of counting.
So you have to find another game. There are two in which you still have the opportunity to beat more than the odds: poker and sports. Even better, the casinos, since they take a small cut of each play, don’t care what mechanisms you use to acquire the extra knowledge you need. The cuts, or vig, they do take do mean, though, that you still have to beat more than the odds.
Sports betting is a world unto itself, where insider knowledge can really pay, usually when betting against somebody acting on sentiment. Figuring the odds in poker given the usual assumptions about a deck of cards, and what cards you can see that have already been played, are easy. Anybody can learn how in an afternoon. Yet even if you master these, you’ll still lose.
You’ll lose because of the casino cut, but you’ll also lose because the expert players, the game’s true psychologists, will beat you. They will beat you because they know how to acquire that needed extra knowledge—you’ll give it to them.
The extra knowledge is written all over your face, in your movements, in the way you fiddle with your chips, in the way you play and bet the cards in front of you, and in the way you played and bet your previous hands.
Poker is huge and growing bigger. The cycle builds on itself: as new players enter, the pros get richer, and those riches entice new player. The cycle repeats. Yet the only way to get good at it is to lose—and learn why you lost.
[At this point, I demonstrate a card-reading experiment, in which I distribute four blank cards and ask people to write their favorite playing card on it. The chance I guess the right person of each card is 1/24. I got all four correct. To the poker tables!]
Addendum for webpage: Oh, that number of all what-are-the-chance questions? It’s 1. The probability of anything that happened is 1. It’s only the unknown, future events that are uncertain. And Monty Hall? The probability on the assumptions the prize is behind door 3 is 2/3, meaning the contestant should always switch.
Thank you for the nice examples of additional information.
Any given casino’s payouts will not align with the “fair” statistical odds, the payouts are always smaller (e.g. a “fair” coin may payout in $.80 increments [or some other increments less than $1.00] when the client wins, but when the client loses the client/patron always loses in full $1.00 increments).
Briggs says, “Probability is always a state of mind.” — but casinos [as one example] apply “probabilities” to ensure, per statistical probabilities [“states of mind”?] they remain consistently profitable. And not only consistently profitable, but profitable in dollar amounts that are very very very closely aligned to predictions based on client/patron betting volume. This raises some very basic questions:
– In who’s mind(s) is(are) the “state(s) of mind” of probabilities residing among those employed by the casino?
(obviously, I’m messing around with this one — what Briggs presents as a “state of mind” he explains as one’s quantity/completeness of information, such as the probability of which of two pockets vs seeing what actually happened [and many statistics courses delve into calculating the value of additional information…calculating estimates of the value of additional information is recurring theme consistently omitted in this blog] — but if he really means “state of mind” in the usual meaning of the words/phrase, I for one would like to what the heck he’s talking about).
– Because casinos do win, consistently, year over year, by applying probabilities to their payouts, does this even qualify as “gambling”?
OBSERVE: The individual outcomes [one bet to the next] reflect unpredictable occurrences [aka, they are “random” — with “random” being a characteristic of the pattern of outcomes [and ‘randomness’ being characterized by a mean, mode, variance, and standard deviation, etc.]; “random” is not some thing or some force causing individual events to occur]. Sometimes the patron wins, sometimes the casino…nobody can know from one bet to the next what the individual outcome will be. However, the aggregate accumulation of the individually unpredictable occurrences is predicted with very high accuracy –casinos know with precision, based on patron gambling patterns, just how much profit they’ll amass well in advance. This high predictability/near-certainty of final outcome argues against the casino’s enterprise qualifying as “gambling.”
– AND because casinos pay out so lopsidedly relative to the actual probabilities, at what point does this lopsided payout become unethical?
The above brings us to the doctrine, held by many religious denominations, that “gambling” is sinful — but precisely what constitutes sinful “gambling”?
The vast majority of the masses of patrons are ignorant of the casino’s manipulations of actual odds vs diminished payouts, is engaging in something about which one is ignorant [a “state of mind”] what makes the sin?:
– is an individual patron, engaged in “gambling” when they play casino’s games? If “yes,” is this always the case, or, only under certain conditions (i.e., does the patron’s “state of mind” matter and if so how/to what extent?)?
Consider the MIT Blackjack Teams that applied systematic techniques, all based on mathematical probabilities associated with a given casino’s rules to gain a statistical advantage over the casino — they actually won at rates well within their expectations. Were the individuals on a given team, or the teams as a whole, “gambling”?
One could argue that because they are applying a system, with disciplined rigor, such that the outcome is predictable with near-certainty, such blackjack teams are not “gambling,” and thus, they are not engaged in the sin of “gambling.”
However, for the casino to operate, and for the blackjack teams to operate, both require masses of ignorant “gamblers” to fuel the casino’s operations — does this impart some sinfulness on both … or… because the blackjack teams are not in any way enabling either the casinos or the ignorant gamblers, are they alone exempt from “sinful” behavior because their system does not constitute “gambling”?
If one accepts that casinos are, by enabling “gambling,” thus also engaged in sinful behavior, and, the blackjack teams are not “gambling” and thus not engaged in sinful behavior … then that suggests a type of context, a “state of mind,” that makes the sin vs no-sin distinction.
That creates a conundrum: If the casino is not “gambling” is its sin one of temptation & enabling other’s [the casino’s patrons] to sin via their [the patron’s] exercise of “free will” (no casino forces anyone to come in and gamble)? Some DO say “yes,” that the casino, by creating the opportunity, is thereby sinning. That is, the creation of a temptation, and means, to sin itself constitutes a sin.
If one buys that, how does that fundamentally differ from God putting the Tree in the middle of the Garden of Eden that tempted A & E to exercise their free will and disobey? If a casino by enabling a sinful behavior is by the criterion of enabling thereby sinning — then didn’t God also sin by creating the same kind of sin-enabling situation?
There’s some very profound, and nuanced, ethical/morality issues here that for which many inquiring minds desire philosophically-based answers on the premise that “gambling” [whatever that is] is sinful (e.g. http://erlc.com/resource-library/articles/a-biblical-case-against-gambling)!
People who don’t gamble seem to always confuse probability and odds Odds gives the pay rate while probability gives your success rate (if correct).
There is a break even probability for any given odds value. You “beat the odds” when your probability of success is accurate and better than this break even value.
Statistic courses assume the break even probability computed from the odds is the actual probability (that is, fair). But your information may be better and thus your probability of success may be as well.
Not sure what “sinning” a has to do with gambling. Everything in life is a gamble. If you drive to work, you are gambling you will get there alive and presumably will profit. When you eat, you are betting your food doesn’t contain a fatal poison.
If the profit of a gamble is mostly assured then the gamble is called an investment. A possible distinction between gambling and investing is: an investor knows (pretty sure anyway) he will profit while a gambler hopes for profit. Casinos invest while many of their patrons gamble.
Contrary to popular opinion casino’s love card counters. There were very few BlackJack tables in Nevada before the publication of the original book on counting “Beat the Dealer.” And after publication, it takes over most of the floor.
Nothing better for the casino than a bus-load of tourists with fat wallets thinking they can beat the casino. Because most can’t. It is not necessarily that hard to count. Most players lack the discipline. To many hours of pure drudgery, playing the basic strategy at the table minimum waiting for the deck to get hot. And even if you do it right, there will be long streaks when luck is against you, and again, few players have the discipline to stick with the strategy when it isn’t working.
One story of a player taking Vegas for $1 million brings several times that back to the city. They just want that $1 million to be extracted from one of their rivals down the strip.
The above brings us to the doctrine, held by many religious denominations, that “gambling” is sinful — but precisely what constitutes sinful “gambling”?
Putting one’s faith in abstract “luck” rather than in a personal God. There is a difference, but usually only understood by those who with a genuinely religious experience. There also is a somewhat fine distinction between “gambling” and the myriad of risks taken as a matter of course and necessity every day. I has to do with motive.
What are the odds of your hotel key card in what pocket would I would guess most of the time it would be the right hand pocket since most people are right handed and it would be awkward to place a card in the left since most people would use their right hand to insert and remove the card with that hand. Now this is only a guess and only true polling with actual showing where such card is placed would tell anyone what is true or not. No computer model based on assumption would be able to predict this, it would only show if the programmer’s assumptions are correct or not that computer model predicted anything.
Correction: On *only* the assumptions provided for the Monty Hall problem, the odds that the prize is behind door #3 are unknown, but are not less than 50%. The conclusion that the odds are specifically 2 out of 3 rests on the added assumption – which nobody ever actually puts in the problem – that Monty not merely *did* reveal an empty bay, but invariably *does* reveal an empty bay. (It’s possible, after all, on your formulation, that Monty didn’t know where the prize was any more than I did, and there was a 1-in-3 chance that, when he opened door #2, the prize would be revealed, the bwah-bwah-bwah music would play, and that would be the end of that. In which case, the choice between #1 and #3 remains dead even.)
And if you reply, in defiance of your own terms, that of course I am supposed to make obvious assumptions based on the nature of television, I reply that, in that case, I might well make the additional assumption that Monty always picks the lower-numbered door wherever possible. In which case, if he had opened #3, the odds that the prize was behind door #2 would be 100%, but, since he opened door #2, the odds that it’s behind door #3 are still 50%. (Why, you ask, would Monty choose a course of action that as good as told the contestant where the prize was one-third of the time? Well, because game-show producers *like* big giveaways. They’re good for ratings; most people prefer to watch big wins rather than big losses.) Or perhaps he has some more subtle tell, which I, as a fan of the show, could deduce after watching a few dozen episodes. Only if his course of action when the contestant picks the prize door is entirely random (relative to his knowledge, at least; tossing a coin before the show would do) are the odds truly 2-to-1 in favor of switching.
None of this, of course, ever makes it *wrong* to switch doors; no matter what the assumption, a consistent switcher ought to at least break even in the long run, and very probably do better. Ergo, if I truly know nothing about Monty’s behavior besides what you told me above, it is quite true that I should always switch. All I wish to clarify here is that the claim to identify the precise odds on the given information is fallacious – and that, if it comes to inventing additional details, I can easily invent quite plausible ones to justify sticking with door #1, should I feel like doing so.