A successful Texas hold em player is one who maximizes his return or minimizes his loss with each decision he makes. Expected value (EV) is essentially the amount one would expect to win or lose if this decision were to be repeated millions of times. (rememer your precalculus "limit as n goes to infinity"??? didn't think so!)
Every poker decision, whether it be to bet, raise, check/call, check/fold, check/raise has an expectation of success or failure. Those that will lose you money in the long run are said to have negative expected value (-EV). Folding always has an expected value of zero. You will never gain or lose money by folding. Folding, though it has an EV of zero, may often be your best decision from an EV perspective.
Let's look at a few examples. Expected value is not always easily calculable, even in limit texas holdem, but every time you play you should try to analyze your decisionmaking with EV in your mind. For instance, you're playing 1/2 and in the small blind with 84o on a flop of K96 rainbow. The small blind checks and only you and a tight player are left. There is three dollars in the pot. Should you bet? You are risking one dollar. Betting has a positive expectation if you can expect the other two to fold more than one in four times. If you try it four times and it succeeds once, you win a three dollar pot and lose your one dollar bet the three other times, making it a 0 EV play. (We're discounting the chance that you are called and catch runner runner or win in some other very unlikely way) Remember, we analyze each decision on its own Expected Value merits. If you are called and catch an 8 on the turn, you must again analyze your decisions based on their likelihood of success or failure in the long run. Poker, and Texas Holdem in particular, is a game of short term variations, but you must continue to make the correct EV decisions and you will be a winner in the long term.
Another example: You are holding A8 of spades, again playing 1/2. There are 3 limpers ahead of you and one behind you. The flop comes K96 with two spades. The first player bets and the other two in front of you call...you should raise! As we've seen in the Poker Odds section, you have about a 35 percent chance of hitting your flush. You will not win each time you hit it...perhaps someone has flopped a set or two pair and will hit a full house, etc. You may even occasionally win if you spike an Ace. Anyway, it is pretty clear that your odds of winning the hand are better than 25 percent, and with 3 players in the pot ahead of you, you want to get more money in the pot while you have an advantage from an expected value standpoint. Now, there are other factors that will enter your head and should be taken into account. If the original bettor reraises you and the other two fold, you've now put 2 dollars in and gotten the others to put 6 in. Now we may or may not be in positive EV territory, depending on what our opponent has. Also, we must consider the fact that our raise may have bought us a free card or may have gotten someone with an Ace and a better kicker or paired side card to fold. As you can see, there are many factors influencing the expected value of our decisions. We may not always be sure we've made the correct poker play, but it's extremely important that our thinking process runs along these lines.
Even preflop you must think along expected value lines. You may be holding AJs in the small blind. Six players, most of them very loose, limp before you. A raise is in order. Think in terms of expected value or pot equity. Your hand figures to win more than one in every seven times against the starting hands held by the others, so a raise is in order. Again, evaluate each decision on its own merits. If you miss the flop in this instance, checking and folding may be your best option from an expectation standpoint. Or, it may not...you may have an overcard, gutshot and back door flush draw...again, do the math in your head and arrive at the correct decision.
Finally, remember this...in general, a bet has a higher expectation than a check/call. You will sometimes win by forcing people to fold. Now, there are times when this is not correct due to the threat of a raise (if you're planning on calling), but always bear it in mind. In Texas Holdem, aggressive poker is winning poker.