{"id":1575,"date":"2013-12-03T13:21:34","date_gmt":"2013-12-03T13:21:34","guid":{"rendered":"http:\/\/beyondnumbers.lvablog.com\/?p=129"},"modified":"2023-06-21T11:30:38","modified_gmt":"2023-06-21T18:30:38","slug":"pushback-and-bayes-rule","status":"publish","type":"post","link":"https:\/\/www.lasvegasadvisor.com\/gambling-with-an-edge\/pushback-and-bayes-rule\/","title":{"rendered":"Pushback and Bayes\u2019 Rule"},"content":{"rendered":"

I\u2019m only one post into this blogging adventure and I\u2019m getting pushback from the cognoscenti. In my intro post, I advised APs to avoid using the word \u201cstuck.\u201d A player then tells me, \u201cUsing that word doesn\u2019t make you a pro\u2014it just makes you old school.\u201d Sigh. I\u2019m not sure where to start. First of all, you don\u2019t 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, \u201cgaming\u201d) 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 \u201cstuck\u201d 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!<\/p>\n

Perhaps a better example of the pushback I\u2019m getting is in regards to the tawdry practice of shuffling chips at the table. Pros all say, \u201cLots of gamblers shuffle their chips.\u201d 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\u2014I must now invoke Bayes\u2019 Rule: P(A|B)=P(B|A)P(A)\/P(B).<\/p>\n

You hoped this blog would never go there, but now look what you\u2019ve done! First, let\u2019s explain all that gibberish notation and derive that formula as the result of simple logic. It\u2019s a bit of a digression, but no trees were harmed in the production of this blog. The P is short for \u201cProbability of\u201d whatever follows in parentheses. A and B refer to events or statements, and the vertical bar | means \u201cgiven that.\u201d So, P(A|B) means \u201cthe probability that event A occurs, given that event B has occurred,\u201d or \u201cthe probability that statement A is true, given that statement B is true.\u201d 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.<\/p>\n

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\u2019m serious\u2014sit 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\u2019s 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\u2019s attach some numbers. If the total surface area of circle B is 10 square inches, and the area of that cat\u2019s-eye intersection is 2 square inches, then the probability P(A|B) is 2\/10 = 0.2 = 20%.<\/p>\n

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\u2019 Rule with pen, paper, and dart!<\/p>\n

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.<\/p>\n

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\u2019 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\u2019re \u201cstuck\u201d?! Good luck, Old School!<\/p>\n

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mgid scam<\/a><\/div>\n\"Facebook\"<\/a>\"twitter\"<\/a>\"youtube\"<\/a>\"instagram\"<\/a>","protected":false},"excerpt":{"rendered":"

I\u2019m only one post into this blogging adventure and I\u2019m getting pushback from the cognoscenti. In my intro post, I advised APs to avoid using the word \u201cstuck.\u201d A player then tells me, \u201cUsing that word doesn\u2019t make you a pro\u2014it just makes you old school.\u201d Sigh. I\u2019m not sure where to start. First of […]<\/p>\n","protected":false},"author":13,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"spay_email":"","jetpack_publicize_message":"","jetpack_is_tweetstorm":false},"categories":[2],"tags":[47,63,65],"yoast_head":"\nPushback and Bayes\u2019 Rule - Gambling With An Edge<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.lasvegasadvisor.com\/gambling-with-an-edge\/pushback-and-bayes-rule\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Pushback and Bayes\u2019 Rule - Gambling With An Edge\" \/>\n<meta property=\"og:description\" content=\"I\u2019m only one post into this blogging adventure and I\u2019m getting pushback from the cognoscenti. In my intro post, I advised APs to avoid using the word \u201cstuck.\u201d A player then tells me, \u201cUsing that word doesn\u2019t make you a pro\u2014it just makes you old school.\u201d Sigh. I\u2019m not sure where to start. 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