Math assigment on Poker, any ideas?

My little sister is about to do a written assignment on her math class. The assignment should be around 15 pages and she is supposed to work on it full time for one week. Her teacher suggested she wrote on probabilities and combinatorics in Poker and my sister found that interesting (runs with the family I guess). She has to come up with an idea for a particular question/formula/topic/idea herself and then her teacher refines that idea into a set of questions.
So my question is: Given a girl at 16 wokring on a 15 page assignment, what could be an interesting topic to write about in relation to poker and math?

[BigBiceps]

One interesting topic could be "mathematically determining if a poker site is dealing non-random river cards in order to influence the outcome to maximize their eventual long-term profit".

Yes, it could be interesting to examine the RNG for the poker sites. There would be enough material to cover 15 pages and you could take it to whatever complexity you like.
However, I would believe this to be too big a task for a 16 year old. The data collection would be hard. Whatever computations on the data, I would guess she does not have the knowledge to interpret the results. The "die hard" test suite I know of to test for randomness is quite complex and it takes a professor to conclude anthing based on the results :-)

[spoohunter]

What about "spots in Texas Hold'em where profit is guarenteed", for example getting all in the money in preflop with AA, getting x amount of money in on the flop with top set, getting money in against a specific hand range, etc, etc.

[yellowjack]

An analysis on assessing the value of all starting hands in all forms of poker. If it's too long, just stick to hold'em.

i.e. "With paired cards, the chances of getting a set on the flop is approximately 1 in 8. In terms of playing the game well, this means that if you are holding a pair and it costs say $1 to call, at least $8 should already be in the pot or $8 should be anticipated in the pot. This is because you will be getting 8 to 1 on for your money in the pot, which is the chance of hitting a set.

To hit a set, there needs to be a 3rd card of that rank in the first 3 cards. There are two 9s left in the deck, so there are C(2,1), or 2, different cards that can come. The remaining cards must be cards that are not of the rank 9, and there are C(48,2) ways for this to happen with the other two cards on the flop. The total number of combinations of flops is C(50,3) and not C(52,3) becuase you are already holding 2 of the cards in the deck. Hence, the probability of hitting a set on the flop is the number of possible flops divided by the total number of flops. In other words, this is:

= C(2,1)*C(48,2) / C(50,3)
= (2/(1!*1!)) * (48!/(46!*2!)) / (50!/(48!*2!))
= 2256/19600
= 0.1151020408

In addition to getting 3 of a kind on the flop in shorthanded pots (i.e. when heads up) pairs can win a showdown unimproved, or without making a 3 of a kind, straight, flush, etc., a large number of times."

From there you could go onto how it depends on how high the pair is, number of people you're against, etc.

You could also go into the probability of one particular hand vs. any other one particular hand, which seems pretty straight-forward and easy to discuss. There isn't too much math in there though. All that comes up there is overs vs. pairs, pairs vs. pairs, two unpaired hands vs. each other (dominated hands, hands like AT vs. KQ, KJ vs. QT, AQ vs. T9).

If you choose to go this way, you have to throw in Gus Hansen's T8o all-in call preflop vs. Esfandiari's 77 in the WPT Bad Boys of Poker!

The power of suitedness could be derived, and that would probably take at least a page if you discuss it thoroughly enough.

After showing starting hand values you could help your sister come up with tournament hands to make analyses on plays when you only know one person's hole cards, a la Harrington, or if you know both of them. You could probably go into assessing a single hand vs. a range of hands and calculate equity. Before you do this, you definitely need to discuss the race situation of overcards vs. pairs.

Going into games like 7-card stud and omaha would be difficult, so you may scratch it.

Anyways if you want more suggestions or some help in coming up with concepts feel free to PM me. Good luck.

[Doc7]

15 pages for a 10th grader (or whatever 16 yr olds are) ????
hot damn is she in genius school?

[pzhon]

[ QUOTE ]
Her teacher suggested she wrote on probabilities and combinatorics in Poker and my sister found that interesting.

[/ QUOTE ]
I suggest writing about probabilities and combinatorics in poker, unlike what many other people have been suggesting in this thread.

How about taking flop situations such as the following:

Flop: 8[img]/images/graemlins/heart.gif[/img] 7[img]/images/graemlins/heart.gif[/img] 4[img]/images/graemlins/diamond.gif[/img]
Player A: 5[img]/images/graemlins/spade.gif[/img] 5[img]/images/graemlins/club.gif[/img]
Player B: A[img]/images/graemlins/heart.gif[/img] K[img]/images/graemlins/heart.gif[/img]

What are the probabilities for wins, losses, and ties?

On the flop, Player B has 15 outs against Player A: Any heart, king, or ace. However, many of those allow Player B to redraw to trips, a straight, or a full house. Also, Player A can pick up 3 more outs if the turn is a 7 or 8, but loses many outs if the turn improves player B to a straight or set.

This situation is complicated and deserved careful analysis, and it does not require any esoteric tools. The goal might be to develop techniques which could be used to analyze other situations with draws and redraws.

One advantage of this topic is that the calculations can be checked against www.TwoDimes.net :

http://twodimes.net/h/?z=1284833
pokenum -h 5s 5c - ah kh -- 8h 7h 4d
Holdem Hi: 990 enumerated boards containing 4d 8h 7h
cards win %win lose %lose tie %tie EV
5s 5c 460 46.46 527 53.23 3 0.30 0.466
Ah Kh 527 53.23 460 46.46 3 0.30 0.534

Thanks for your input everyone! I have printed them out and will talk with my little sister and see if she can gain some concensus from her teacher, as it is ultimately a job between them to come up with a feasible yet fun exercise.

And for the comment about my sister being a genius. :-) At her school (not elite) every student has to make a written assignment on 15 pages. They will get a week free from school to do it. They can choose any topic in history, danish, english, math, physics and so on. In order for the students not the cheat, they decide on the topic, but the ultimate questions to be answered by them will be given at the start of the week.

I went to the same school 10 years ago. I wrote about Galileo, Copernicus and Newton as a history project. I wish I had the option of writing on poker. [img]/images/graemlins/smile.gif[/img]