I’m only one post into this blogging adventure and I’m getting pushback from the cognoscenti. In my intro post, I advised APs to avoid using the word “stuck.” A player then tells me, “Using that word doesn’t make you a pro—it just makes you old school.” Sigh. I’m not sure where to start. First of all, you don’t want to be old school! You want to be an idiot tourist, one of the millions of new gamblers who only recently discovered the wonders of the Pairs Plus when gambling (er, “gaming”) was finally legalized in their state last year. You want to be the guy who boasts to his friends that he now has been to Vegas 6 times, oblivious to the fact that knowing the exact number undermines the spirit of his boast. On top of that, guess who defended “stuck” as an acceptable word in a casino? None other than a longtime professional player who is a member of the Blackjack Hall of Fame! QED!
Perhaps a better example of the pushback I’m getting is in regards to the tawdry practice of shuffling chips at the table. Pros all say, “Lots of gamblers shuffle their chips.” The only ones who defend the practice are the habitual abusers. I agree that lots of gamblers shuffle their chips, but as a percentage, way more pros do. If you abusers are going to deny the correctness of my advice, I have no choice—I must now invoke Bayes’ Rule: P(A|B)=P(B|A)P(A)/P(B).
You hoped this blog would never go there, but now look what you’ve done! First, let’s explain all that gibberish notation and derive that formula as the result of simple logic. It’s a bit of a digression, but no trees were harmed in the production of this blog. The P is short for “Probability of” whatever follows in parentheses. A and B refer to events or statements, and the vertical bar | means “given that.” So, P(A|B) means “the probability that event A occurs, given that event B has occurred,” or “the probability that statement A is true, given that statement B is true.” In our example, we want P(Pro|Shuffle), that is, the probability that a person is a professional player, given that he shuffles his chips.
To derive the general formula, draw two partially overlapping circles on a piece of paper. Label the left circle A and the right circle B. (I’m serious—sit down and do this!) Suppose I tack the paper to the wall and throw a dart at it, and then tell you that my dart landed in circle B. I ask you: What is the probability that the dart is also in circle A? I am asking you to compute P(A|B), the probability that the dart is in circle A, given that we know it is in circle B. Since I told you the dart landed in circle B, the only way it can also be in circle A is if the dart landed in the intersection where the circles overlap (the piece shaped like a cat’s pupil). So, the probability P(A|B) is a simple fraction: the area of the overlapped intersecting piece A & B, divided by the total area of circle B. Let’s attach some numbers. If the total surface area of circle B is 10 square inches, and the area of that cat’s-eye intersection is 2 square inches, then the probability P(A|B) is 2/10 = 0.2 = 20%.
We just gave a graphical example of the principle that P(A|B)=P(A & B)/P(B). If you actually sat down with a piece of paper, this description was very simple and enlightening. If you are confused now, it means that you ignored my instructions, and took the lazy route of hoping that the blog discussion would be a sufficient explanation. (No one listens!) If my dart had landed in circle A, we could compute P(B|A), and by the same reasoning, P(B|A)=P(A & B)/P(A). By substituting out P(A & B) in these two equations (I will explain some simple stats, but NOT algebra!), we obtain P(A|B)=P(B|A)P(A)/P(B). We just derived the legendary Bayes’ Rule with pen, paper, and dart!
Now back to our story, we see that P(Pro|Shuffle)=P(Shuffle|Pro)*P(Pro)/P(Shuffle). From here, I think we should just attach some numbers to illustrate by example. Suppose we have 1000 players, 10 of whom are pros. Assume 80% of pros shuffle their chips, but only 10% of civilians do (I think these are reasonable numbers, except 1% of players being pros is way too high). So, we have 8 shufflers out of the 10 pros (1% of the 1000 players), and 99 shufflers out of the 990 civilians. In all, we have 107 shufflers out of the 1000 players, a fraction of 0.107. In the absence of any information about you, the boss would say that the probability that you are a pro is 1%. She uses only the fraction from the overall population to give her a probability.
Now she sees you shuffling chips, so she wants to know P(Pro|Shuffle), the probability that you are a pro, given that you shuffle your chips. The intuitive way to apply Bayes’ Rule is to think of what cases are excluded by the given information. Here, seeing you shuffle means that you are one of the 107 chip shufflers. The other 893 players are excluded. The chance you are a pro is thus 8/107, since 8 of the 107 chip shufflers are the pros. We can apply the formula by plugging in: P(Pro|Shuffle) = P(Shuffle|Pro)*P(Pro)/P(Shuffle) = 80% * 1%/0.107 = 0.0748. By shuffling your chips, you just raised your probability of being a pro in her eyes from 1% to 7.48%. And on top of that, she hears you say that you’re “stuck”?! Good luck, Old School!
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Funny, Bayes’ Theorem has always been touted as some kind of worldshakingly profound concept, but to me, it’s always–even when I was a kid–seemed trivially obvious. I didn’t even need to draw circles to understand it.
Of course, that doesn’t mean that the thickheaded majority understands it. In one significant consequence of mathematical illiteracy and general stupidity, the likelihood of a false positive in something like, say, a drug test that is 95% accurate is grossly underestimated by not just the unwashed masses but by the very supposedly learned people who administer and evaluate such tests.
To return to gambling, I prefer games where I don’t have to conceal the fact that I know what I’m doing, such as video poker, sports betting, or live poker. (I realize that there is a necessary element of camouflage in each (“Mister, which horsie do you think is gonna win?”), but it isn’t nearly as mandatory as in, say, card counting–though I was a successful counter for eight years with no cover whatsoever, and yes, I did use the word “stuck” at least 8,715 times.)
Excuse me while I run out and buy a new hat, because according to Kevin I apparently have a very thickhead. Oh wait, can I afford it? Why yes I can, because I make money gambling.
I am retired military. James’s note brings to mind what was taught to us about not speaking of classified information even in “talk around.” On the surface it may seem giving little, insignificant pieces of personal information is harmless but when they start adding them together they begin to reveal more than was initially intended.
While in general I agree with you on chip shuffling, in my opinion a large % of people who shuffle chips at a pit table are poker players also. Now, we can have the discussion over whether someone who plays a lot of poker is more or less likely to be a counter, (I’d say slightly more), but at a minimum, proficiency at chip shuffling makes someone MORE likely to be a standard degen, for the very reasons stated. Personally, I handle chips horribly, and therefore am constantly playing with them at the table, stacking and restacking them, as if there is some rhyme or reason to my constantly-changing bet sizes related to my stack size.
I love it. Pieced-together information has been the downfall of many operations, gambling and otherwise. I could tell some stories.
It’s a rare pit boss who understands Bayes’ Theorem. The typical pit boss’ thinking is going to be more along the lines of, “A lot of professionals shuffle their chips. Therefore, if I see a player shuffling his chips, he’s probably a professional.” One of the lessons of Bayes’ Theorem is that since there are many, many times as many ploppies as pros, if you see a player shuffling chips, the odds are still heavily against him being a pro. The real danger is that most pit bosses aren’t going to understand this.
I’m making the point Bayes’ Rule shows that despite the counter’s rationalization that “lots of gamblers shuffle chips [or say certain words],” Bayes’ Rule tells us that the probability of being a pro goes way up. You’re making the point that despite the pit boss’s belief that “all APs shuffle chips,” Bayes’ Rule tells us that the probability of being a pro is still very low. These are both true, and they create a “Heads-I-Win-Tails-You-Lose” scenario. By shuffling chips, you legitimately raise your “pro probability,” and on top of that, the boss’s ignorance raises this pro probability still further. Just stop shuffling and talking like a pro!
When playing BJ I often find myself reflexively doing a simple small-stack chip shuffle;
BUT I always make sure to do it so badly that the chip stacks tumble more often than not.
Just a minuscule bit of camouflage. Nothing more.