Doing the maths

This material is taken from Roy Brindley´s website with he´s permission

Did you know there are 1,326 possible [2-card] starting hands in Hold’em? Or 2,598,960 possible five-card hands in poker? Read on for more statistics on poker but first some of the basics…

HAND RANKINGS

Royal Flush: Five card sequence, from 10 to the Ace in the same suit. (eg. 10,J,Q,K,A)
A royal flush is a combination of a flush and a straight ending in the Ace high card. So all the cards are of the same suit, consecutive and have the Ace high card.

Straight Flush: Any five card sequence in the same suit. (eg. 8,9,10,J,Q and A, 2,3,4,5 of same suit).
A straight flush is a combination of a flush and a straight. So all the cards are of the same suit, and all are consecutive. Ranking between straights is determined by the value of the high end of the straight. A royal flush is a straight flush that has a high card value of an Ace.

Four of a Kind: All four cards of the same index (eg. K,K,K,K).
Four cards of the same value such as four jacks or four 7's represent the second strongest poker hand. This hand beats everything except a Straight Flush.

Full House: Three of a kind combined with a pair (eg. A,A,A,5,5).
A full house is a combination of three of a kind and a pair. Meaning all five of your cards are a part of a set of either two or three of the same card value (eg. three 7's and two Kings). Ties on a full house are broken by the three of a kind, as you cannot have two equal sets of three of a kind in any single deck.

Flush: Any five cards of the same suit, but not in sequence.
A flush is a hand where all of the cards are the same suit, if each card you have is all one suit, such as 3 of Clubs, 5 of Clubs, 6 of Clubs, 8 of Clubs and King of Clubs, then you have a Flush. Don't be tricked into thinking that all five cards are the same color. The high card determines the winner if two people have a flush.

Straight: Five cards in sequence, but not in the same suit.
A straight is a hand where all of the cards are consecutive. There is no continuative quality to this poker hand a straight cannot wrap around meaning it is not a straight if you have a Queen, King, Ace, Two or Three. Standard poker rules state that in the case of more than one straight, the higher straight wins, In case of straights that tie, the pot is split.

Three of a Kind: Three cards of the same value.
Any three cards with the same value (eg. a 6 of Clubs, a 6 of Spades or a 6 of Diamonds) is considered to be three of a kind. The highest set of three cards wins.

Two Pair: Two separate pairs (eg. 4,4,Q,Q).
Two sets of two cards of equal value constitute a hand that has two pairs. As usual the pair with the higher value is used to determine the winner of a tie.

Pair:
One pair of two equal value cards constitutes a pair.

High Card:
When the hand you are left with has no pairs, is not a straight or a flush then it's relative value is determined by the highest value card. When two players have no pairs, straight, or flush the winner of the tie is determined by the highest value card in the hand. If the highest cards are a tie then the tie is broken by the second highest card. Suits are not used to break ties.

CHANCES OF THE HIGHEST CARD IN YOUR HAND PAIRING AND IT BECOMING TOP-PAIR ON THAT FLOP.

CHANCES OF AN OVERCARD ARRIVING ON THE TURN OR RIVER

If you get top pair from the flop, how likely is it that you will get an over-card on the turn or river?

ODDS OF RUNNING INTO THESE BETTER HANDS THAN YOU PRE-FLOP

ODDS THAT YOU WILL BE DEALT AN ACE

COUNTING OUTS

For those new to the game, an 'out' is a card that will make or dramatically improve your hand. For example, if you are holding 10c, Jc and the flop comes 2d, Ks, Ah, you have four 'outs' that will make you a 'nut' straight: Qh, Qs, Qd, Qc.

Here is a more complex hand: You are holding Ad, 10d. The flop comes 10s, 7d, 2d. With this example any diamond will give you a 'nut' flush and any 10 will give you 'top-trips' which will also be a tough nut to crack. There are naturally nine Diamonds left in the deck plus a 10c and 10h, meaning, in this case scenario, you have eleven 'outs'.

Now, each 'out' has a mathematical chance of arriving and often your job as a player is to asses the cost of a call at that point (after the flop) against potential profitability. We are now verging on 'pot odds', something even more complex so, instead, here is a quick-reference way of calculating those percentage chances.

Put plainly, each 'out' when there are two cards to come - the turn and river cards - has an approximate 4% chance of arriving. That is for the first ten 'outs' and then each one thereafter equates to an additional 3%. With just one card to come - the river - each 'out' then boats an approximate 2.2% chance of arrival.

So, one of your eleven 'outs' [looking back to our Ad, 10d situation], has about a 43% chance of arriving when there are two cards to come and, with one card to come, just 24.2%.

Like I say, these figures are 'rules of thumb' close approximations. Let's face it, if and odd 0.2% makes a difference to you, you deserve to win the World Series! Trust me, you will get the hang of these maths very quickly, because it's simple: Every out represents 4% for the first ten cards then 3% for each thereafter. With one card to come, each out represents 2.2%. What could be easier?