Poker Maths: all your statistics and odds

[Twigman]

Right, I have been meaning to do this for a while, and was considering a blog, but this way others can contribute as well.
I have an A level in maths and an additional AS-level in Statistics. I feel this qualifies me sufficiently to talk about odds and the maths of poker.

Hopefully people can ask questions and I should be able to answer them.

Some basic odds for starters:
Basic facts you need to know (and lets face it should know if you are playing poker).

• There are 52 cards in a deck, therefore the odds of drawing one from a complete deck is 1 in 52 (1/52 = 0.0192, which is about ~2%).
• There are 4 suits of 13 cards each, the odds of getting a card in a specific suit from a complete deck is 13 in 52 (13/52 = 0.2500 25%)
• There are 4 cards of each value (one of each suit), the odds of getting a certain value card is 4/52 (1/13 = 0.0769 ~8%

Preflop
Pair: You want a pair of the same value card. You are dealt your first card so the probability is 1. You don't care what the value is. The second card must be the same as that card. There are 51 cards remaining, and 3 are of the correct value. This is 3/51 and for completeness we should multiply by the 1 as we want a card and a matching card so 3/51 x 1 = 0.0588 or ~6%. This means you will get a pair once every 17 hands ish.

Aces: You need to get an Ace (4/52) followed by another ace (3/51) = 0.0045 ~0.5% This is about 1 in 220 hands (this is the same for any specific pair). This is also a 13th of the value for any pair as there are 13 different pairs you can get.

Suited Ace King: You need any king or ace (8/52) and then the suited other card to make the hand (1/51) = 0.0030 ~0.3% Once every 340 hands.

Suited anything: Any card, followed by a card in the same suit (12/51) = 0.2353 ~24%

The probability of me getting Aces OR Suited King Ace is 0.0045+0.003 = 0.0075 0.75% Once in 130 hands.

I'll do flop odds probably tomorrow.

If people want more info on just general statistics and how it works I can do this to. This is not meant to be a one of post. I plan on adding to it. And feel free to post as well and please correct me if I'm wrong...I hope I'm right. I'll look stupid if I'm not! Ah well just trying to help

[Pates]

Great post twiggy. These will be awesome odds to refer to in the future. +rep

[thefairy]

i will add big time to this tomorow but i cant be assed to get supersystem out of the case and do all the apendixes lol.

[James]

Nice one twig. Nick and i have discussed it and we are adding +15 to your competition post count for the competition for this post.

[Twigman]

Winner I just tried to +rep you for that but need to spread it around. I'm fully planning on doing Flop odds and other things as well, it's just a bit late tonight. I would have done more, but had to take Christmas tree down as well.

[SirJonnyP]

Nice one! It'll be good for some of the fishy players here to understand some odds. Tan.

[Twigman]

Flop
(ignoring hole cards at the moment)
Pair: 1 (any card) x 3/51 (a matching value pair) x 48/50 (not a match) = 0.0564 ~6% or about 1 in 18 flops *Note this is incorrect, see below for correction*

Three of a kind: 1 x 3/51 x 2/50 = 0.0024 0.24% 1 in 425 flops.

Three of the same suited cards: 1 x 12/51 x 11/50 = 0.0518 ~5% 1 in 19.

Rainbow flop: 1 x 39/51 x 26/50 = 0.3976 ~40% 1 in 2.5

So once you know this you can start doing things like combining this with your hole cards.
Say you hold a pair, what's the probability of
flopping a set with pocket pair? (ignore probability of getting dealt a pair in the first place)

The card that makes trips can be the first in the flop:
2/52 * 50/51 * 49/50 = 0.0369
Second in the flop:
50/52 * 2/51 * 49/50 = 0.0369
Or third in the flop:
50/52 * 49/51 * 2/50 = 0.0369
Note how all the odds are the same. It is equally likely you will make trips on any card in the flop (as they all come out at the same time).
You need the probability of any of these happening so add them up and you get 0.1109 11% chance to make trips or 1 in 9.

Four of a kind with pocket pairs
2/52 * 1/51 * 1 = 0.0007
2/52 * 50/51 * 1/50 = 0.0007
50/52 * 2/51 * 1/50 = 0.0007
Any of these so add them up = 0.0023 0.23% 1 in 442
This means if you are dealt pocket pairs, the chance of you flopping 4 of a kind is 1 in 442 flops.

As before correct me if I'm wrong etc. And any questions please ask. More coming soon

[Pates]

Great post Twig. I think there might be something wrong with calculating 1 pair flopping. I thought 6% was too low.

I worked it out as

3/51 that the 2nd card on the flop pairs the 1st card. Then 6/50 that the 3rd card pairs either the 1st or the 2nd (unpaired 48/51) card. So 3/51 + (48/51 x 6/50). Which gives around 17%

I really don't know if that's right at all though.

[ccottis]

Yeah the odds of flopping a pair part you've done is the odds of the first two cards pairing (and not getting trips)
Actual odd will be 3* this (as you can pair 1st and 2nd, 1st and 3rd or 2nd and third)

[GarrethUoL]

Great post! This will definitely come in handy in the future, especially on the "is poker rigged?" debate!