On Tuesday, I am making a 45-minute presentation to the Math Club at the University where I teach English. I'm sure I'm trying to do way too much and leaving out every single important nuance, but here's what I'm planning to cover. Pot Odds
The ratio of the money in the pot to the call you're making should exceed the odds against your getting the card you want. If you have AsQc and the flop is 9s7d3c, you have six pretty clean outs (3 aces and 3 queens). If there is $2.00 in the pot, and someone bets $0.50, you are getting 5:1 to call. You should fold (if you believe you're behind). On the other hand, if there were $4.00 in the pot and someone bets the same $0.50, you're getting 9:1 and should call. If you have AsKs and the flop is JsTs4c, you have 18 (9 spades, 3 queens, 3 aces, and 3 kings) pretty clean outs (assuming no one flopped a set—even then you have 12 pretty clean outs). You're likely behind right now; however, you're the odds-on favorite to win. You can call any bet; in fact, an all-in shove is not a bad play.
Going back to the first AQ example above, you don't really have odds to call. However, if you believe that your opponent will pay you off if the A or Q comes, then you have what is called implied odds. For example, if you believe that, if you get your A or Q on the turn, your opponent will call your bet of $1.50, then you are betting your $0.50 with the chance of winning $4.00, not just $2.50. Now you have correct odds to call. In No-Limit Holdem, you'll rarely actually have pot odds to call; it is a game of implied odds. The trick is to know how your opponents are likely to behave. If you habitually overestimate your implied odds, you'll go broke.
Middle to low pocket pairs will rarely win unimproved. When you play them, you are hoping to “flop a set” and win a big hand. The odds against this are 7.5:1. The vast majority of the time, you're going to miss your set. You must be willing to let your pocket pairs go after you miss the flop. Moreover, when you do hit your set, you must make enough money to make up for the 88% of the time you're losing your preflop bet. Therefore, you must be sure that both you AND your opponent have about 10 times the bet you're being asked to make before calling preflop with low or middle pocket pairs. For example, if your opponent raises to $1.5 preflop, you need to make sure that both you and he have at least $15 left before you call this with a low pocket pair. The math for playing suited connectors (9s8s) preflop is almost exactly the same as the math for playing pocket pairs.
If you are head-up on the flop against a single opponent, there's a good chance that you both missed the flop. Assuming that neither of you has a pocket pair, there are 6 cards out of 50 that give you a pair, so you'll hit about 36% of the time, as will your opponent. You'll both miss 28%+. If you were the preflop aggressor, you should bet most of the time on the flop because your opponent has missed the flop 64%+. Even if he flopped bottom or middle pair, he may not be all that keen to defend his hand against the strength you're projecting. Let's say that you raise to to $1.50 in a .25/.50 game. Everyone except the big blind folds. There is now $3.25 in the pot. The big blind checks to you. You totally missed the flop, but bet $1.50 anyway. For the sake of argument, let's say that you lose every single time he calls, but don't put any more money in the pot. When he calls, you lose $1.5 x 0.685 = - 1.0275 When he folds, you win $3.25 x 0.315 = +1.0237 In other words, if he calls only 2/3 of the time, you win money. This is the most elemental level of game theory. You want to bet the right amount of money and the right percentage of the time to make it impossible for your opponent to make the right decision here. The more you bet, the more likely he is to fold, but the more risk you're taking. If you bet the pot, $3.25, he needs only call 50% of the time to defeat your edge. If you bet only $1, he needs to call 76.5% of the time to defeat this edge. Most c-bets are in the 1/3 to ½ pot range, big enough to get folds, but not big enough to represent real risk. If you show preflop strength, you can sometimes get away with c-betting two opponents. If you c-bet as a bluff against three or more opponents, you're throwing money away.
River Bets / Calls
The river card has been turned. Your hand is what it is. You can no longer draw to a better hand, nor can your opponent. One of you has the better hand. Pot odds and implied odds no longer exist. A completely new kind of math takes over. Let's say the pot is $10, and the river card was significant (an ace, the third of a suit, a card that makes a possible straight). Alas, it didn't help your hand, which at this point, you know is a loser. However, you make a $5 bet anyway, pretending that it helped your hand. When he calls, you lose $5 x 2/3 = $3.33 When he folds, you win $10 x 1/3 = $3.33 In other words, if you think you're opponent will fold to your ½ pot bet over 1/3 of the time, you're making a profit with this play Turning it around, you're now on the calling end of this play, there is $15 in the pot, and you're being asked to call $5. You call. When you win, you win $15 x ¼ = 3.75 When you lose, you lose $5 x ¾ = 3.75 If you think your hand will win ¼ of the time or more, you should make this call. My friend Travis hates it when I say, “I only had to be right 20% of the time to make that call.” He counters, “It was only one hand and you were either going to win or lose it.” He's right. But I'm going to play many hands in my career. Just as importantly, my friend Travis knows that I'm not automatically going to let him have the pot on the river just because he bet at it.
A long time ago, David Sklansky developed what he called the “Fundamental Theory of Poker.” In essence, this is it: “make bets that make your opponents make mistakes.” When they call your bet, you want them risking more money than the odds would dictate. When you're ahead, this is often a pretty small bet. For example, if you have top pair and your opponent has middle pair and no flush or straight draw, he has five outs (2 cards to a set and 3 cards to 2 pair). He needs 8.2:1 pot odds to call you. If you bet $0.50 into a $2.00 pot, and he calls you, he's making a mistake. (Assuming you won't pay him off if he hits—see “implied odds” above). However, the situation will rarely be this simple. Let's say, you still have top pair and he still has middle pair, but there are two of the same suit on the flop. Now he may have 14 outs (the five before, plus the 9 flush cards). With 14 outs, he can reraise you all-in and be the odds on favorite to win by the river. In fact, if you bet at all when you're opponent has 14 outs, you're making a mistake. And, of course, you just can't know. For this reason, most of the time, you should bet somewhere between 1/3 of the pot and ¾ of the pot when you think you're ahead. You want to entice your opponent into risking more than his hand is worth by calling, without doing the same yourself. If you bet only 1/3 of the pot, you're giving straight and flush draws the wrong odds to call (barely) but not combination draws. If the turn or river card gives your opponent the possible straight or flush, you have to be able not to pay him off; if you do, you were giving him the correct implied odds to call. Now, knowing whether or not he's bluffing is another matter altogether—see game theory above.
What is your target audience? Are you assuming that your audience knows how to play poker, or not? If there is a chance that some do not know how to play poker (even if they think they do), then you may need to give a basic introduction.
Your examples don't really have the correct poker information. AQ and AK could easily be the best hand, while you are counting them as draws. The implied odds section incorrectly says that you will rarely have the pot-odds to call; you often do, when you are calling because you think you have the best made hand, or when you have a combination draw. The set-mining example is incorrect, since you often lose or do not get paid off when you hit a set, so you need much better implied odds than 10:1 to call for set value alone. The continuation bet example assumes you lose if you check behind, which is wrong. Your river example included pot odds, even though you said it didn't.
As a mathematician, I'd present different material than you would, but I suggest you think about what you want to accomplish in this presentation a lot more. You aren't going to teach anyone how to play poker well.
I suggest taking some simple examples, and nail them. You might try something like the Sklansky Chubukov numbers, or a simple model of bluffing, or using the independent chip model in a SNG.