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.