I was a losing player five years ago who had dropped $200 in buy-ins. Joined FTR, turned it around with a $100 buy-in and Poker Tracker + a HUD. That was back when you bought the HUD separately. I cannot thank FTR enough for the five years of advice and encouragement. I'm a hobby player and a mathematics professor with a teaching load that consists of Game Theory and 4 different probability and statistics courses. Lifetime in poker, I've made back all the money I donked off to start with plus all the money I spent on software and books (nearly $500 worth, as I use some of them for background/examples for my classes), plus an additional $2,500.
I'm a math geek and an underachieving poker dumbass. Ask a mod. They won't disagree. I'm happy with my poker life, though I get accused of "mathturbation" regularly. I get to play a game I love and, every now and again, get to withdraw some money. Just back after a year-long, post-Black-Friday layoff.
For the special 3k occasion, let's play "stump the math geek." I'm going to post some probabilities that I think about at the tables. You don't have to know these factoids to be successful at poker, but some of them help a good bit. Then I'll take questions. Ask me the probability of anything you like, and I'll take a stab at it. I'll do better with gambling related probabilities, but I will attempt any problem you post.
I'll use spoilers to hide the maths, so the post is less scary. The first spoiler is just some notation we'll use in later spoiler-math sections.
Spoiler:
We'll need factorials like 4! = 4 * 3 * 2 * 1 = 24. Note that 0! = 1. We'll need choose notation for the binomial coefficients, so let's go with 5c2 = 5! / ( 2! * 3! ) = 10. This is all the ways to choose 2 objects from a collection of 5 objects where order does not matter, e.g. AK = KA. I'll leave the maths in spoilers so the reading's not so daunting. Check my maths if you're of a mind to.
Example 1: When playing HU, I love a 2 pair hand - bottom two, top two, top and bottom - love 'em all. You can often get all the chips in with 97 on a A97 against any Ax hand. And it's usually the nuts. Hero flops exactly two pair (when starting with unpaired cards) about 2.02% of the time, or just slightly more than once every 50 hands. This should also be an argument AGAINST playing hands like 73o in any game, since the likelihood of a big flop with it is about 1 in 50.
Spoiler:
Let 2PrF be the outcome of Hero starting with two unpaired cards and flopping exactly two pair, e.g. one of each card Hero holds. (We're ignoring times when we flop one pair and the board pairs). Then:
P(2PrF) = 3c1 * 3c1 * 44c1 / 50c3 = 3 * 3 * 44 / 19600 = 396/19600 ~ 2.02%
Explanation: suppose Hero has 97, the "3c1" expressions choose exactly one 9 (and one 7) from the three remaining 9's (and 7's), and the "44c1" chooses any one of the remaining 44 non-9's and non-7's.
Example 2: I love paired boards. I view them as "inflection points" when analyzing a villain. How you play paired boards tells me how good your poker awareness is. Instead of hitting the flop 1/3 of the time, a Villain playing a hand like AJ+ will only hit a paired board 1/5 of the time. And when he hits, he has either a big hand or a one pair hand knowing there's almost no chance we've hit something. You can see why paired boards are gold mines for reads on postflop play. When do they barrel? Do they recognize my donk into them with an underpair for what it is, just the mathematical knowledge they can't have hit the flop very often? Donk/fold is easy, as is bet/fold in position. When they hit, they hit hard and know they're way ahead. So they usually let us know, too. This gives a great deal of insight into how they play top pair type hands on other board textures. Same is true for overpairs they hold - they know they're likely way ahead and usually can value bet with impunity. We get a chance to recognize it for exactly what it is and note that for later.
2a) In general, we like playing hands like AK because they flop a pair or better about 1/3 of the time. The exact probability is 32.43%, with 2.02% two pair, 1.35% trips, .09% full houses, and .01% quads. That leaves 28.96% one pair hands that are TPTK. Villains like playing them to, and even the brain dead villains know to cbet. When I see cbet > 75%, I'm looking for a way to extract value from all those times he's barreling with nothing but overs.
Spoiler:
Assuming we are using Hero's hole cards to make pairs, etc, there are 6 cards that improve our hand. The overall probability of hitting at least one of them:
P = ( 6c1 * 44c2 + 6c2 * 44c1 + 6c3 * 44c0 ) / 50c3 = ( 5676 + 660 + 20 ) / 19600 = 6356 / 19600 ~ 32.43%.
The break down is:
- exactly one pair: 5676 / 19600 (~28.96%)
- exactly two pair: 396 / 19600 (~2.02%)
- exactly trips: 264 / 19600 (~1.35%)
- exactly a full house: 18 / 19600 (.09%)
- exactly 4-of-a-kind: 2 / 19600 (.01%)
Explanation: the "6c1" picks one card to pair up with one of our while the "44c2" picks any two other cards - the "6c2" picks any two cards, giving us our two pair hands and trips - the "6c3" picks all three flop cards to match one of ours, so these are boats and quads.
2b) How does this change when the board pairs? If we assume Villain has unpaired cards to begin, he has only a 20.43% chance of hitting the flop. We can get a read on his barreling since we know that only 1/5 of the time he has a hand, and the rest of the time he's got air. Also remember that more than 2/3's of the time the board pairs (depending upon his and our hole cards) it pairs with T's or lower, another problem for top pair hands. They're pretty sure we don't have anything, but they KNOW they don't have anything and are unlikely to improve.
Spoiler:
We proceed in two parts and assume Villain is holding two unpaired cards. First, we count all paired board flops that MISS him. Second, we count all the possible paired boards flops that HIT him.
Part I: To count all the flops that DO NOT hit either of Villain's two hole cards:
Combos = 11c1 * 4c2 * 40c1 = 11 * 6 * 10 * 4 = 2640
Explanation: choose one of the 11 card values NOT in Villain's hand with "11c1", then "4c2" to get a pair; choose any one of the 40 remaining unused cards with "40c1"
Part II: To count all the flops that HIT at least one of Villain's hole cards:
Trips = 2c1 * 3c2 * 44c1 = 2 * 3 * 44 = 264
1 Pair + Board pair = 2c1 * 3c1 * 11c1 * 4c2 = 2 * 3 * 11 * 6 = 396
Full House = 2c1 * 3c2 * 3c1 = 2 * 3 * 3 = 18
Explanation (Trips): pick one of Hero's two hole cards with "2c1", then count the trips with "3c2", then pick any of the 44 remaining unused cards with "44c1"
Explanation (1 Pair + Board Pair): pick one of Hero's hole cards to pair with "2c1", then make the pair with "3c1"; choose any of the remaining 11 card values for the board pair with "11c1", then make the pair with "4c2"
Explanation (Full House): pick one of Hero's hole cards to be the trips with "2c1, then make the trips with with "3c2"; we've already selected the card to pair by NOT selecting it to be trips, so just make the pair with "3c1"
Note: there is an easy double-counting / half-counting mistake. If interested, I will explain in the thread below. The difference is easy to see when dealing 5 cards and counting all the possible two pair hands compared with counting all the possible Full Houses.
Summarizing, there are 2640 + 264 + 396 + 18 = 3318 paired boards, and Villain hits 264 + 396 + 18 = 678 of them, or ~20%.
Example 3: You know how you hate those flush draw boards? Get used to 'em. If we're observing a table, and know none of the hole cards, we would see the flop come with 2-of-a-suit about 55% of the time. Thinking about playing 98s? How often can you expect a flush draw in your suit? About 11% of the time. Additionally, about 1% of the time we flop the flush immediately.
Spoiler:
Flush Draw Flop: 4c1 * 13c2 * 39c1 / 52c3 = 4 * 78 * 39 / 22100 = 12168 / 22100 ~ 55.06%
Explanation: The "4c1" chooses the flush draw suit, with "11c2" ways to pick 2 flush cards; then "39c1" ways to pick any other card
Hero hits flush draw on flop when holding suited hole cards: 11c2 * 39c1 / 50c3 = 55 * 39 /19600 = 2145 / 19600 ~ 10.94%
Explanation: The "11c2" picks two cards from Hero's suit, with "39c1" picking any other card
Hero hits flush on flop: 11c3 / 50c3 = 165 / 19600 ~ .84%
Explanation: The "11c3" picks three cards from Hero's suit, with nothing else needed
Note that we use 52c3 = 22100 when counting total possible flops with no hole cards known, and 50c3 = 19600 when the hole cards are known
Now, it's your turn. Ask me anything. What poker probabilities have you wondered about but never felt like calculating? Or any probability problem that interests. Ask me, and I'll do my best.
Good luck at the tables!
