math for 169 types of starting holdem hands

I've been re-reading my probability books for the math we're doing with ofc. I thought it would be good to look at some hold'em stuff too.

How does the math show that there are 169 TYPES of starting hands in texas hold'em?

SHORT ANSWER

There are 13 types of pairs.

Ignoring suits, there are C(13,2) types on non-pairs. That comes to 13!/11!2! or 13*12/2 or 78. Factoring in suits, we have 78 suited non-pairs and 78 non-suited non-pairs.

13 types of pairs
78 types of suited non-pairs
78 types of non-suited non-pairs
-----------------------------------
169 TYPES of starting hands

LONG ANSWER

There are C(52,2) starting hands. That comes to 52!/49!2! or 52*51/2 or 1,326.

Of course this counts A A and A A as two different hands so we need to do some consolidating to get to 169 TYPES of starting hands. We know that each of the 13 pairs can be made C(4,2) or 4!/2!/2! or 4*3/2 or 6 ways so there are 13*6 or 78 pairs.

Here are the 6 pairs of aces:
A A
A A
A A
A A
A A
A A

We can treat all 6 of these pairs of aces the same. Consolidating, we have 78/6 or 13 TYPES of pairs.

That leaves 1,326 - 78 or 1,248 non-pairs.

Let's look at AK.

There are 4 ways to have AK suited:
A K
A K
A K
A K

There are 12 ways to have AK offsuit:
A K
A K
A K
A K
A K
A K

K A
K A
K A
K A
K A
K A

This means we have 3 times as many offsuit hands as suited hands for the 1,248 non-pairs. Let x equal suited non-paris. x + 3x = 1,248 which means 4x = 1,248 which means x = 1,248/4 which means we have 312 suited non-pairs. That leaves 936 offsuit non-pairs.

Consolidating the 312 suited hands, we know we can treat each of the 4 suits the same so we divide by 4. This means we have 312/4 or 78 TYPES of suited hands.

Consolidating the 936 offsuit non-pairs, we know we can treat all 12 of the above AK offsuit hands the same. This is the same for all 936 hands so we have 936/12 or 78 TYPES of offsuit non-pairs.

We're left with the same totals as the short answer:
13 types of pairs
78 types of suited non-pairs
78 types of non-suited non-pairs
-----------------------------------
169 TYPES of starting hands

As you can see, it is more likely to get some of the 169 TYPES of hands than others. AK offsuit is more likely than AA which is more likely than AK suited.

*Note that we used combinations and not permutations because we're not concerned about the order in which the two hole cards were dealt. In other words, we treat A 3 the same whether we were dealt the ace first or the trey first.

Without wishing to be rude, isn't the short answer 13*13?

Originally Posted by The Bean Counter

Without wishing to be rude, isn't the short answer 13*13?

I think 13*13 happens to be right because XYs is not statistically equal to XYo, AND the paired combinations do not have a "suited" form.

Both facts make a 13 by 13 chart a convenient tool, but if XY had other than 2 important forms (suited and offsuit being our 2), OR IF the pairs came in more than one form (offsuit is ours), then the simplification of 13*13 is invalid.

You can instantly see that if the suits had a ranking to each other, then A3 would NOT be equivalent to A3... If we allow >>>, where the suit of the highest kicker would be a 'tie-breaker' if these hands were otherwise equal against each other at showdown, there would be a slight statistical advantage to the hand with the A. (EDIT: since the 3 is far less likely to be a playable kicker.)

It is the way we subdivide 1,326 into statistically equivalent groups that leads to the final number of significantly different hands.

Thanks for the explanation and makes sense. OP really is what my old boss would refer to as a "sledgehammer to crack a nut" given what we are trying to calculate, but I'm sure the above can be easily adapted for calculating different hand groups for different games if you have a decent level of mathematical ability (although worryingly, these calcs are miles clear of anything British kids learn at school).

No it's not. Perms & Combs are A-level Maths S1 module on most exam boards.

The idea's touched upon on most GCSE syllabuses. I was looking at a question last week that said you go to a restaurant and they are doing a deal where you get starter & main or main & dessert.

Starters
prawn cocktail
pate

Mains
Lamb
Chicken
Fish

Pudding
Ice Cream
Chocolate Cake

How many possible combinations are there you can have?
How many combinations contain Ice cream?

Which is definitely the idea behind the whole concept it just doesn't waste time going into more complicated notation or harder examples which would confuse most people.

Yeah, definitely the notation that confuses the shit out of kids with maths. That and terrible teachers, which is why a lot of kids don't go on to A Level Maths.

I think I did something like that for my GCSE coursework actually. Iirc, top marks were given if you worked out the nCr formula by yourself (piece of piss obviously).

In Soviet Russia math figures you out.

Haha I am so confused! Interesting though

Related: http://www.flopturnriver.com/pokerfo...rs-165910.html

Scroll down in that post to make sure you see it all since it's over multiple posts.

Originally Posted by spoonitnow

Related: http://www.flopturnriver.com/pokerfo...rs-165910.html

Scroll down in that post to make sure you see it all since it's over multiple posts.

You are so sick, was literally doing the same thing myself so cheers for saving me time.

No problem. I made that post almost five years ago. God I'm old.

Edit: Shit, the original post I meant to link to was http://www.flopturnriver.com/pokerfo...tc-161721.html not the one above

I'm glad you linked the wrong one, can already do all of that basic crap (thanks to reading that post).

Originally Posted by The Bean Counter

Yeah, definitely the notation that confuses the shit out of kids with maths. That and terrible teachers, which is why a lot of kids don't go on to A Level Maths.

Wow, this is incredibly offensive and ignorant. I'm sorry to derail the thread, but I can't let this go unanswered. The idea that a sizeable proportion of bad teachers are personally to blame for kids leaving school without a grasp of simple mathematics seems to be a favorite of Michael Gove at the moment. It makes a convenient scapegoat for underfunding of schools (and huge class sizes), poverty among students and their families, a testing system set up to disengage kids entering schools at a low level of education, and a government set curriculum and inspection system that deprives teachers of any flexibility in teaching method/content anyway.

The majority of teachers work incredibly hard to cope with overfilled classes of kids who are often not at all interested in even being there and who have often not been exposed to books/science/maths or intelligent conversation until the day they arrive.

I went to a school like this, which was even threatened with being shut down while I was there. For most of my final year at the school we had a police van on one school gate and a police car on another at the end of each school day. We even had a police helicopter overhead more than once. This school was the definition of a problem school, and yet, less than 4 years before it had been one of the better schools in the area. That all changed when another underfunded trouble school nearby was closed, and our class sizes doubled overnight with the influx of new kids.

Despite all of this, I ended up getting a masters in physics at university and a fairly good grasp of basic mathematics along with that. I had exactly the same teachers as the rest of these kids. Trying to claim that a group of people who work incredibly hard to educate (often quite troubled) kids are on average bad at their jobs is not only incredibly insulting and wrong, but also damaging. If you actually want to see schools get better then stop blaming teachers and start fighting for more money to flow into education. Teachers are fucking heroes working some of the longest and hardest hours of any job in the country and they deserve our respect and support.

*Mod: please delete this derailment post in due course*

I agree that good teachers are worth their weight in gold. They get a terrible press in the UK and I’m one of their biggest defenders, largely because I get to see first hand the commitment required from somebody who is an Outstanding teacher as judged by OFSTED. Mrs Bean Counter works 70+ hours per week during term time and quite often in the holidays (including in some of the roughest schools in Birmingham until recently) and is under continual pressure from the state and Joe Public.

I also agree that teachers are deemed responsible for a number of problems and my remark was somewhat flippant. I think the main issue is around bad parenting and particularly parents that are looking to absolve all responsibility for the educational and social upbringing of their own children. Teachers cannot be deemed responsible for the lack of literacy in the UK and they certainly cannot be blamed for lack of manners or any number of other social issues (some primary school children still arrive at school wearing nappies for instance). It’s no secret that kids with academically motivated parents far outperform others at school and in life, be it through reading to their kids, checking if they have done their homework, promoting the virtues of education, etc. I worry about a member of my family in particular in this respect – this poor little kid wasn’t taken to his first day at school because his biological father couldn’t be bothered to get out of bed. I’ve also heard this kid saying “daddy says there’s no point in going to school”. Frightening.

One area that teachers can have a direct impact on however is by creating a learning culture that celebrates success (for the hours the kids are inside the school at least). That’s not to say parents aren’t still incredibly important in this respect,but Outstanding teachers by definition are extremely creative and deliver lessons that are engaging to nearly all students (which is essentially the point I was making). They also use classroom management where appropriate to prevent others disrupting the learning environment and ensure that all students are making sufficient progress in each and every lesson. In essence, the attitude that being a teacher in a disadvantaged area is a damage limitation exercise is largely an excuse. Likewise, a number of teachers in privileged areas are getting away with it to an extent, since the kids will achieve no matter what the standard of teaching. The latter is currently a problem Mrs Bean is experiencing in her new school, particularly as she is not allowed to view more than one in every 250 lessons that her staff deliver

It certainly sounds like we went to the same school. My secondary school was in the worst 1% in the country the year I completed my GCSEs in terms of pretty much every measure you can imagine (results, teaching quality, etc). Teachers and pupils were assaulted on a regular basis, the majority of male pupils carried a knife to school and drug taking was pretty common. Like you too, my group of mates and I have all gone to become professionals and all achieved top grades throughout education. That said, it would be dangerous to assume that our GCSE teachers must have therefore been good teachers (irrespective of the number of hours they work). My mates and I all did well in spite of our terrible teachers because we are all Gifted in teaching terms and have academically motivated parents. Otherwise, how would students like us get the top grade in Maths for instance, when our two teachers were either “photocopying” for half the lesson before returning smelling of booze or whose idea of teaching was to open our textbooks and do some questions? The reality is that smart kids will pass their GCSE’s for the simple fact that they are ridiculously easy but that’s no good for the lesser ability kids that want to do the best that they can.

I don’t know the detail within Michael Gove’s plans, but Mrs Bean has mixed opinions. She certainly isn’t impressed with the lack of clarity around the curriculum and the ridiculous lack of time that will be allowed to prepare for next year (once the Govt has got its act together). That said, she is pro- performance-related pay and anybody striking against it will be a “who’s who?” of Poor or Satisfactory teachers that hate change. I don’t support any teacher striking on this basis and the fact that Mrs Bean is paid less than each of the six teachers she manages and outperforms simply for the reason that they have been in the profession longer is absurd. Bear in mind that there isn’t an awful lot she can do to help teachers move up the pay scales either, since this is a profession whereby she cannot view more than 1 in every 250 lessons her team teach due to Union rules. As to why a teacher wouldn’t want to be reviewed more to be the best teacher they can possibly be is beyond me, but this is a ridiculous system only in place in education that needs to end if we want a step change in teaching quality in the UK.

Finally, despite what the popular press will tell you, teachers are also largely left to it. Yes, they have to deliver certain elements of the curriculum, but they can deliver that however they wish unsupervised.

Anyway, I hate derailing a thread and happy to discuss over PM if you wish.

It sounds like your full position is more reasonable (except supporting an obvious attack on the salaries of all teachers under the name of "performance related pay").

That being the case, you should be very careful that your flippant comments don't support ongoing attacks on teachers pay and working conditions when you don't really mean it.

You do it again by implying that a sizeable proportion of your teachers were alcoholics. I have no doubt that there are some teachers who suck at teaching and even some who sneak off in the middle of a lesson to get pissed, but I can't believe that this is anything other than a tiny minority. Both of your posts imply otherwise. There is a battle going on right now in education, and every flippant comment you make that supports the idea that a sizeable portion of teachers are "terrible" makes it that little bit less likely for the government to see any resistance to a program of further slashing of school budgets, forcing teachers to work (even) longer hours for lower pay, and extending teacher retirement age until they drop dead on the job.

We can get back to the original topic now.