Odds, Outs and Pot Odds
As much as poker is a game of instinct, character, reading your opponents and, yes, even luck, it is also a game based deeply in mathematics and probability. It can be no coincidence that many of the best exponents of the game in the world today have a keen understanding of the mathematics required to make critical decisions in poker and the likely outcome of certain situations.
There are some key probabilities with which you should become acquainted if you want to give yourself a big edge over your opponents. Have a look at this Poker Probabilities chart and you'll see the percentage chances of certain events occurring or hands appearing.

If you are determined to wait until you get Pocket Aces before you play then you should know that the odds of that happening are 220-1 against, or 0.45%. Likewise, if you want to bide your time holding on for a premium hand like A-A, K-K, Q-Q, J-J or A-K suited, then it helps to know this will only occur 2.1% of the time. If you're someone who likes to play two suited cards in pursuit of a flush then you should know that those two suited hole cards will typically only go onto to complete a flush 6.5% of the time.

Understanding the chances of an event occurring will help you greatly in your decision-making process, so take the time to know the likelihood of certain situations occurring. You can try to defy the mathematical odds if you like but over a sustained period of time they balance out fairly evenly.

Having looked at probability, we should also familiarise ourselves with outs. These are vitally important to know when you're in the heat of the action and need to be aware of what you can hit to make your hand and the odds which exist of that happening.
Poker is always played with fifty-two cards and so we are able to make predictions on what cards may appear, based upon the knowledge we have gained by those cards we can see. If we have been dealt A-K as our starting hand in a game of Texas Hold'em then we know there are fifty cards we still haven't seen, of which six are the other three Kings and three Aces. Therefore, if we want our A-K to make top pair on the flop then we know that there are six more cards which can help us. That means we have six outs, and there are five cards to come on the flop, turn and river. With this knowledge we can begin to work out the chances we have of making our hands.

If that flop comes without an Ace or a King then we've now seen five cards in total and that means there are forty-seven we haven't seen, of which six are the cards we need. That's roughly a one-in-eight chance, though we have two cards in which to get it (the turn and the river), which gives us a one-in-four chance of making top pair.

If it all sounds complicated then don't worry, it really isn't that bad. Most of the time you will be able to easily work out how many outs you have and what the chances are that you'll hit them. For instance, if you have 7-8 in your hand and the flop is 5-6-A then you have an open-ended straight draw. Either a 4 or a 9 will make your straight. There are four 4's in the deck and also four 9's and so you have eight outs in total. There are eight cards for you to catch on the turn or the river to complete your hand.
If you follow this link Outs & Odds you will see that we have provided a chart detailing the number of outs and the chances that a player will hit them. It makes for informative reading and you should acquaint yourself with it.

You won't always have access to that chart so it's handy to know how to work out your outs and odds. An easy way to work out the chances of you to catching the outs you need after the flop is to multiply them by four. It is not an exact science but it is close enough to give you a good idea of your chances. Going back to that open ended straight draw, if you multiply your eight outs by four you have thirty-two. A look at the chart shows the actual chances to be 31.5%, which is close enough.

Another example would be if you were holding K-K but suspected that you'd need to make three Kings to win the hand. There are two Kings left in the deck, so if you multiply that number by four then you get eight. Another quick look at the chart will show that the exact chances are 8.4%. Again, that's close enough.

If there is only the river to come then you simply multiply your number of outs by two, to get a figure that's approximate enough for you to make your key decision.
Once you're armed with all of the information regarding odds and outs you need to be able to use it to help decide which plays are profitable and whether or not you should be betting in certain situations.

There are times when pot odds can be used to show that you HAVE to call, regardless of the strength of your hand. Suppose you were in a tournament and the blinds were 3,000 and 6,000. You are in the Big Blind and have posted the 6,000. A player in mid-position goes all in for 6,500 and another player calls that by posting 6,500. The small blind now also calls the extra 3,500 and the action is now on you.

Including your original big blind of 6,000 there is now 25,500 in the pot and all it is going to cost you to see the flop is another 500. You are therefore getting odds of over 50-1 on your extra 500 and MUST call without hesitation, regardless of what cards you're holding. It doesn't even matter if you've got 2-7 off-suit because there's no way you're going to be anything worse than a 10-1 underdog, yet you're getting 50-1 to call. That's pot odds of more than 5-1 in your favour! If you got lucky and flopped a full house you'd probably go on to win a pot of at least 26,000 and all for an extra bet of just 500.

That is an extreme example of what can happen but the principle always holds true. If you're always getting positive pot odds when you play, provided that you're playing with decent hands, then eventually you're going to be making money.

To give another example; Suppose you'd just seen the turn card and there was now 12,000 chips in the pot, with you still needing a card to make your flush. You can now work out where you stand and whether you should continue to draw for your flush.

Your opponent decides to go all in for their last 2,000 chips. That means there is 14,000 in the pot and you have to put in another 2,000 to see the river. That's 7-1 for your money. If you are drawing to a flush then you have nine outs and we can see by the outs and odds table this means you have a 19.6% chance of getting your card. That's approximately a 4-1 shot and considering that you are getting odds of 7-1 on your money you can see that you really have to make that call.

On the flip side of the coin, if somebody puts you all-in and you're only getting, say, 2-1 on your money then you should not be gambling if you know you've only got a 10% chance of winning the hand. 2-1 for your money on a 9-1 chance is NOT a good way to play poker, or any gambling game for that matter!

Also, if you're absolutely convinced that your opponent already has a powerful hand that you probably can't beat then no favourable pot odds should entice you to call. Good poker players will work out the right sort of raise to try and tempt you to call, because the pot odds are right, and that's where your poker instinct should tell you that you're being strung along or slow-played.

In the long run though, making the appropriate calls based on outs, odds and pot odds will work in your favour, as will folding your cards when you have a marginal hand and no good odds to call.

There are times in tournament play when you don't want to risk your chips, such as when you're protecting a small stack and don't wish to gamble, regardless of the favourable odds (perhaps when you're close to the pay-out structure?), but at all other times you should pay heed to them and act accordingly.
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