12-to-1 shot misses

[peterbirks]

So, Gutshot loses. I thought that their narrow line of argument (that poker is a game of skill and therefore not covered by the act) was likely to be on dangerous ground when faced with the mathematical abilities of your average group of twelve good men and true.

Al Alvarez was on Radio Four's news last night, and although he rubbished the verdict, he failed to observe the simple principle that the concept of "chance" when applied to a single event makes no difference to the concept of "near certainty" when applied to 10,000 events. In other words, the 1968 law itself was logically meaningless.

This is not helped when Nic Szeremata blithly informed the judge that the chance of being dealt a pair in the hole was one in thirteen.

OK, let's give Nic the benefit of the doubt. He was in the jury box, a pressure-point at the best of times, and mouth might have engaged before brain. Then again, he also described "gambling" as "betting at unfavourable odds", which is an interesting concept, but presumably means that if I offer you 4-to-6 on a coin toss, and you take the bet, backing heads for six grand, and the coin comes up heads, I can walk away four grand poorer, pleased with myself that I wasn't gambling when I took the bet.

But then our good Mr Galloway pointed out the "12-to-1" error on the Gutshot forum in this thread: http://www.gutshot.com/bforum/showthread.php?t=20205

And just read a few of the responses. Three of the respondents continue to swear that the chances are 12-to-1 against.

It kind of reinforces one's belief in one's own game. If there are people who consider themselves poker players out there who have so little understanding of the basic mathematics of poker, surely there must still be a future.

Simon, I couldn't be bothered to register to post, but I think that the simplest way to explain it to these people is to take the whole deck. (This is how Scarne does it in Scarne On Cards. It's mathematically inelegant, but it enables the non-mathematical to see the truth of the probability.)

There are 52 x 51 possible two-card combinations = 2652 combinations. But this counts As Ad as different from Ad As. Therefore there are 1,326 combinations when the order does not matter.

There are six possibilities of AA. As AH, As Ad, As Ac, Ah Ad, Ah Ac, Ad Ac, when the order in which the cards appear does not matter.

There are thirteen ranks in the deck, from AA down to 22.

Therefore there are 78 (6 x 13) possible pairs as starting hands when the order does not matter.

So the chance of a pair in the hole is 78/1326. This equals 1/17.


Perhaps then they might see it.

Then again, do we want them to?

Comments

Combination therapy

[(Anonymous)]

There is an even simpler derivation. Whatever card you are first dealt, there are 3 left in the remaining pack of 51 that make your pair. 3/51 = 1/17.

Re: Combination therapy

[peterbirks.livejournal.com]

You need to read the thread (you obviously haven't) to see why this derivation was too complex for the three posters on that thread.

I am of course aware of the line you mention -- it's the one that all poker players use. It was the line used by Simon to correct the initial assertion.

It didn't work. Hence my use of the slow, inelegant and "unsimple" explanation.

As with most mathematically litereate people, you are confusing "elegant" with "simple". The elegant mathematical explanation is rarely the easiest to understand. For that you need slowness, and inelegance.

See The Mathematics of Poker for many examples of how the complex can be explained to the mathematically untrained. It isn't "simple" and it isn't "elegant" (and it frequently requires repetition), but it's needed for those who haven't studied probability or for those who do not understand it intuitively.

It's no surprise that these guys are players rather than teachers. I've yet to meet a mathematics teacher who was good at mathematics who could teach mathematics. The best teachers of mathematics were those who were not very good at the subject, but understood it to a relatively basic level.

PJ

Re: Combination therapy

[(Anonymous)]

OK understood. You are correct, I haven't read the thread. In fact I can't as I am at work and amusingly the site is blocked "Reason: Gambling". Lawyers, who needs 'em.

Re: Combination therapy

[peterbirks.livejournal.com]

You are forgiven! I too am blocked from the gambling sites at work now.

This has odd impacts. I can access the top of Wintermute's comments, but not the threads, and while "Get It Quietly" is ok, "Secrets of the Amateurs" is not.

And while I am banned from betting a fiver on the Australia-England game on Betfair, I am allowed to punt a grand or two on Finspreads.

Bring on Derek Kelly to cite that one....

PJ

Re: Combination therapy

[simong-uk.livejournal.com]

I don't think that way is going to look any easier Pete! I posted again with an example flop of 7,7,7 and also something similar to the first comment on here, but I'm not going to try any harder than that.

FWIW my degree was in stats - I have never used it in the day job but the probability (which was actually A level) comes in useful of course. It allows the more vanilla calculations to be done at the table (open end straight or flush draw, filling up 2pair, etc). When I get in an argument with some scrote I normally offer a simple mathematical proof to support. When quizzed why they believe so strongly their answer the reply normally is "coz I read it somewhere." Bless em.

A simple trick is to calculate the number of desirable outcomes over the number of possible outcomes, using + or * for OR or AND.

So the probability of dealing a royal flush in 5 cards would be:

20/52 * 4/51 * 3/50 * 2/49 * 1/48

Obviously no use at the tables, but a flush draw on the end normally has 9/46 chance, for example, or a shade over 4/1 etc etc. Grandmother, eggs, but others I know struggle with how to approach the basic maths.

Re: Combination therapy

[peterbirks.livejournal.com]

I agree that it doesn't look any easier, Simon, but in my experience the easiest way to convince the denyers of obvious probabilistic truths is (if possible), to show them all possible events, then to show them all the events which they are talking about, and then to divide one into the other. Even better, you ask THEM to divide one into the other. This seems to convince them that it is they who have arrived at the correct answer, and makes it less embarrassing for them to change their stance.

I admit that it may not work to convince people so well on paper....


PJ

Re: Combination therapy

[simong-uk.livejournal.com]

You are more sympathetic to their embarrassment than me :) I find usually the only way to get them to shut up is to try and get them to put decent money on it.

Re: Combination therapy

[geoffchall.livejournal.com]

I guess Pete, that you've been working with odds calculations and in particular, multiplying odds together since betting shop days. I have vague recollections of you going on about how to shortcut the method necessary for working out yankees and other strangely-named bets.

The effct of all this is that your best-guessing ability on odds situations is usualy worked out on instinct before you ever turn round and try and justify with actual maths.

Re: Combination therapy

[simong-uk.livejournal.com]

Pascal's Triangle would be my route...

http://hsb.iitm.ac.in/~jm/ARCHIVES/Mar-April02/articles/math4.jpg

Re: Combination therapy

[real-aardvark.livejournal.com]

Beer:Martini Winterbier and Pripps Byggeria Arboga. (Well, you do it with music.)

Yup, I thought of Pascal's triangle too, largely because it's what A-Level Maths (whatever that was: I believe it no longer exists in any meaninful sense) teaches you to think of when dealing either with combinations or permutations.

But, like your solution, it's way too hard, and completely non-intuitive. Which is the key. We may well live in the age of the calculator, but a quick five minutes hanging around the till at Tesco's will convince you that very few people are comfortable even adding things up, let alone multiplying fractions. Yes, your solution is good and fine. It'll make the poor jurors drop off their perch with their eyes crossed, though.

The (legally speaking)sensible answer is indeed the anonymous one at the top of the thread. Apart from anything else, it can be demonstrated physically, which is more than can be said for dividing 1326 by 78. Unless you have an awful lot of matches, or twiglets, or something in your pocket at the time.

The 3-in-51 demonstration wins on two counts.

(1) It's basically induction, which makes it a step-by-step process. Got a pair? Handle the first card, then deal with the consequences. Humans are pretty much hard-wired to think this way.
(2) You can present it as a (rather shabby) magic trick, if you like. Get the judge, or the foreman, or even opposing counsel to "pick a card. Any card." After that it's easy to demonstrate that the odds of matching that card are 3 out of 51; and, crucially, you didn't do any of the work -- the punter did.

I wouldn't call this elegant. It's just straight-forward pattern matching, and as such is indeed trivially simple. I guess the gimmick would lie in the presentation.

Well, I was lucky in that all bar one of my Maths teachers at school were very good mathematicians and even better teachers. My Dad was a decent enough mathematician and an incredibly good teacher, from all the comments I've heard. 'Course, I was privileged to go to a Direct Grant grammar school, the likes of which have disappeared down the shit-brown brick road along with A-Level Maths and a number of other marks of civilisation no longer considered necessary to the operation of "Cool Britannia." BTW, WTF was Blair thinking? The use of the adjective "cool" in this sense had been dead as a doornail since the early seventies. I realise that the then prevalent "bad" might not work so well in marketing terms, but at least it would have saved 55 million people (minus the odd stoned hippie) from cringing for the next ten years...

Yes, most great mathematicians are lousy teachers. That's why they're locked away in the research department, or occasionally in a merchant bank somewhere. My experience (and it's wider than most people's) of merely "very good" mathematicians is that they are generally very good teachers. After all, in essence, Maths is very easy to teach until you get to the postgraduate level. It's easy for the teacher to get enthusiastic about, for example, deriving the concept of simple harmonic motion from scratch. All you need is an enthusiastic student...

... and that, generally speaking, is the problem.

Cf also Richard Feynman, a true genius and a superb teacher, when he forced himself to be one. Buy the books. They really are eye-openers.

I love this idea of "it's not gambling unless the odds are against you." This is an inadvertent defence of casinos everywhere, the more so because these days I understand they take away the only realistic chance you have of beating them, ie to have significantly more money that the House and to keep betting it until you win. Not a great marketing line for casinos, of course, but it wouldn't surprise me to hear New Labour coming out with it some time soon.

And I've just hit the cretinous maximum of 4300 characters Will the wonders of LiveJournal never cease?

Re: Combination therapy

[real-aardvark.livejournal.com]

Yes, Doubleday, you're right. LiveJournal's posting limit really sucks. By the way, did you notice the "allowable html" stuff below, which of course only comes up when you go through "More Options..." (I'll have Assisted Suicide, waiter, please. On someone else. To go.) and then "Preview".

Oh dear. Committees proliferate, and it is truly camel-breeding season in the wacky world of the Internet. Anyway, Doubleday, what was the rest of your comment? I've forgotten where it started now. But no matter. Cut and Paste is a superb basis for coding Internet sites (that and Cargo Cult Programming, which I am sure is responsible for the magic number of 4300), and, luckily, it also allows me to present you with the following residual gem:

Anyway, so, basically, I think that Geoff is right. You see the multiplication of probabilities as simple, because that's what you're used to. Perhaps I see the 3/51 thing as simple, because that's what I'm used to. I suppose you could offer the reasonable argument that with a Venn diagram of possible jurors out there who are (a) betting shop habituees, (b) darts players -- still an amazing facility for number, albeit not multiplication per se -- and (c) accountants -- although they seem to make up the numbers as they go along, so perhaps they don't count -- the disjoint set of candidates who do not fit any of these categories is very small indeed.

On the other hand, I can't help wondering how Mr Szeremata feels whenever he steps up to the poker table now. I mean, fucking embarrassing or what? And I don't buy the pressure cooker argument: this is just plain ignorance, as demonstrated by the three posters who still didn't get it, even when it was demonstrated in front of their faces.

Could we start a web-site dedicated specifically to this particular bet, and invite them in?

[andy-ward-uk.livejournal.com]

Overall, I don't think Gutshot were well advised here. I can't agree with anyone who disses the verdict on technical grounds. They were obviously in breach of the law. If you argue that the law is an ass, that's another matter.

Andy.