Poker Odds

Hi BSHers and MMBEers,
Here's a fabulous write up on poker and odds provided by Devi Huggins.
Thanks, Devi!

The odds of a hand in a game

Poker is played with 52 cards, 4 suits(hearts, diamonds, clubs, spades), 13 cards in a suit (ace through king, and ace can be highest or lowest)


Hands in the dame(ranked in the order best to worst)

Royal Flush = (Ace, King, Queen, Jack, 10 – all of the same suit)

Straight Flush = (5 cards in numerical order, all the same suit, no 10 as that makes it a Royal Flush. Ace can be a low card here)

Four of a kind = (4 cards of the same numerical value, and another random card. )

Full house = (3 cards of the same number and 2 cards of the same number, but not the same as the 3)

Flush = (5 cards of the same suit)

Straight

Three of a kind

Two pair

One pair

High card



Odds of Choosing 5 cards out of 52:

- There are 52 ways of choosing the first card.
- 51 ways of choosing the second card
- 50 ways of choosing the third card
- 49 ways of choosing the fourth card
- 48 ways of choosing the fifth card

The total number of choices then are: 52*51*50*49*48 = 311,875, 299

Since the 5 cards can be arranged in any order, they can be arranged in 5*4*3*2= 120

different ways, the number of ways of choosing 5 cards becomes

(52*51*50*49*48)/( 5*4*3*2) .= 311,875, 299/120

The probability is written as52 choose 5 or 52C5 = 2598960

(XCY = X!/(X-Y)!/Y!

Odds of a Royal Flush:
There are 4 Royal Flushes, one for each suit. So the odds are 4 out of 2598960

Odds for Three of a kind:
There are 4 ways to get a triple out of 4 suits. and 13 possible numbers, so there are 13*4=52 possible triples.

There are 2 remaining cards to choose (5 cards in a hand) . They cannot be the same rank as the triple (it would then become a 4 of a kind), which means there are 48 cards remaining to choose from. Also the 2 cards cannot have the same rank (as that would make it a double and make the hand a full house) so there are 44 choices for the last card.

and 2 ways to arrange the 2 cards. So the number of combinations for the last 2 are 48*44/2 = 1056.

Total possible hands for three-of-a-kind is then (13*4=52)*(48*44/2=1056) = 54912.

The odds of getting this hand is 54912 out of 2598960.


Odds for Four of a kind
There are 13 ways to select 4 cards of the same number. There are 48 cards left to choose the last cards – that is 48 ways.

So the number of choices are 13*48 = 624

The odds of getting this hand is 624 out of 2598960


Odds of a Full House

This is a triple and a double –

odds for a triple= (13*4 = 52 )

odds for a double = select 2 (same number) from 4 (different suits) = 4C2 = 4*3*2 /2*2 = 6

52*6 = 312

Odds for a full house then is 312 out of 2598960

Odds for a Straight Flush
That is 5 cards in sequential order, all from the same suit. The lowest card cannot be a 10, as that will make it a Royal Flush.

There are 9 ways of choosing 5 sequential cards, if we exclude the card with the lowest as 10.

Since there are 4 suits, the number of choices are 9*4 = 36.

The odds of this choice is 36 out of 2598960.