Summary of special online bonuses

The concepts of expected value and odds are absolutely fundamental in poker.

What is expected value?
Let's consider the following situation: you are playing no limit hold'em, heads up against a player whom you know will, for whatever reason, go all in preflop, all of a sudden (immediate all in raise) with any pair comprised between sixes and tens.
You are in small blind (0.50$) and get .
You both have 50$ stacks.
You raise to 3$. The big blind shoves all in.
You know he doesn't have a very strong hand and you only need to spike an ace to win the pot. There's that little chance to hit a flush too.
Then you call.
Your opponent shows , confirming your past observations on his play.
Board reads .......
You pocket a nice 100$ pot.
In fact, the way you played your hand, you lost money!
A hand evaluator (i'll put mine online as soon as i've finished it) would tell you that, on 100 times you will face that situation, you would, in average, win 33.5 times, lose 66 times and split the pot an additionnal 0.5 time (for instance, whenever the board is all spades).
On 100 hands, on average, you would have won 33.5 x (100$ pot - 47 of your call) + 0.5 x (0.5x100-47) - 66 x 47= -1,191$.
On the long run, it's like you lost 1,191/100=11.91$ every single time you called his all in reraise!
In comparison, if you opt to lay down your hand, you don't lose anything.
If you want to know your overall loss on that hand (each time you're dealt and your opponent has a pair between sixes and tens), then it 3$ if you raise to 3$ and then fold (because now you're sure he has the best of it) while it's 14.91$ if you raise and don't have the discipline to fold your hand to his reraise.

This situation can be compared to heads or tail game.
A friend of yours proposes you this game: he tosses the coin, if it's heads, you pay him 2$, and if it's tail, he pays you 1$. Of course, you will refuse to play with him, without necessarily understand the concept of expected value. You will just think: we have the same probability to win, so we should win/risk the same.
This is the concept of odds in poker: so that you accept to play a hand, make a call etc... the amount of money that you may win must be linked to your chance of winning.
For the heads and tail game, if you play with a fake coin that will go heads twice more often that tail, that won't be an issue. You only have to make sure your opponent will pay you twice more when you win than when he wins.
If you get 2.5 odds on your wager; then playing the game will clearly show a profit for you on the long run. Your opponent can be lucky and start with 5 heads in a row but, if you can play as long as you wish then you will make money at the end (an average 0.17$ win per toss).
How did i get this 0.17$ figure?
It's really easy. It's the expected value of the variable "My win if i play the heads and tail game". This is a random variable. You can't predict its value. You may play one time and lose, this variable will be equal to -1$. You play again and win, it's equal to +2.5$.
The expected value of this variable is its average. It's the constant x defined in such a way that, for both players, each of the two following statements are equivalent on the long run:

1.In a game where player A has a 1 third probability to win, A receives 2.5$ when he wins, and pays 1$ to player B when he loses.
2. Whatever the result of the game, A is paid directly a constant x by B.

For a game with a finite number of possibilities (clearly true here as only 2 outcomes are possible: A wins or B wins), the expected value is just the sum, on all the possible outcomes, of the product "probability of outcome" x "value of variable when this outcome occurs".
Here, it gives us:
Expected value of A's win per toss = 1/3 (=probability for A to win) x 2,5 + 2/3 x (-1)=0.17$.


And as concerns poker?
Being a good player is sticking to hands that have a positive expectation value and playing hands in a way that maximizes their expectation value.

For instance, in NLHE, if you know that, whatever the amount you will raie preflop, your opponent will call, then go all in preflop with aces is the strategy that will maximize the expected value of this hand.
Now, if you know that, whatever his hand, your opponent will shove preflop in an almost empty pot, then the optimal strategy when you are dealt a mediocre hand like is to lay it down immediatly (don't even call the blind).


How will you make the right decision in a more realistic situation?

First, you have to estimate the hand of your opponent or at least know your situation: do you need to improve to win the pot or do you already have the best hand?
Then, take into account the odd of your hand to win and the duo (profit in case of victory, loss in case of defeat). This duo will determine what we call pot odds.

Example:
Preflop, UTG raises to 4$.
Two callers.
You are on the button and look at and decide to call.
At the start of the hand, all players have 100$ stacks.
The raiser is a quite predictable player. He raises very few hands in this early position. So you put him on a big pair.
You call the raise, blinds fold behind you.
Flop is , Pot=17.5$, 4 players
It goes check ... check
Turn=
Raiser fires a 15$ bet.
1 call, 1 fold.
You feel that, with 2 opponents, in a pot raised by a not very aggressive player, there's little chance you can win that pot unimproved.
However, you feel that the pot will be yours if you make your flush on the river.
As in heads or tail game with the fake hand, you have to determine two things to decide whether you make the call or not: your probability to win and the duo (profit in case of vcitory, loss in case of defeat).
9 cards in the deck will give you a flush. In poker, people will say you have 9 "outs".
There are 52-4=46 unknown cards.
So you have 9 chances out of 46 to make your flush on the river. You will win 1 time out 5.1 (46/9).
You will call if there is at least 4.1 times more to win than to lose in this hand. What you will lose if you don't make your flush, it's those 15$ you need to pay to see the river (if you don't make your flush by the river, you don't bet anything more).
What you are "sure" to win if you catch a club, it's 17.5+15+15=47.5$.
However, there should be at least a 4.1x15=61.5$ reward for your call to make it worth it on the long run.
So, apparently, you should lay down your hand.
The pot odds (=size of the pot/price to pay) are inferior to the odds of your hand (=number of times you will lose/number of times you will win). Here: 47.5/15=3.2 to 1 < 4.1 to 1.
But, if you make your flush on the river, you have a chance to extract an extra bet from your opponents. The raiser, with not that much experience under his belt, might find it very difficult to lay down his big pair to your reasonably sized river bet, wanting to "check" you really have him beat.
The first caller might be on straight draw (for instance with QJ) and the card that makes your flush might make his straight too. You could earn a lot of money on a river like or .
On the river, there will be a 62.5$ pot if you call. You can plan on extracting another 20$ from the raiser and a lot more from the other player if he makes his straight.
So you can, in the pot odds calculation, include the future amount you plan on extracting if you make your hand. If you think you will win a minimum of 47.5+20=67.5$ if you make your flush, then the pot will lay you an implied odd of 67.5/15=4.5 >4.1.
Implied pot odds= include future bets that you will extract if you make your hand
In these conditions, you have to call the 15$ turn bet.On the long run, it will win you money (out of 100 times, you will win 19.6 times 67.5$ and will lose 80.4 times 15$, for a total profit of 117$, which makes a net profit of 1.2$ per call).
Another parameter has to be taken into account: you might very well lose even if you make your flush!
If the raiser is far less predictable and can have .
Then you will only win if the river is a club other than or , which gives you only 7 outs now.
The odds of your hand are only 39/7=5,6 to 1 now ! (= you will lose 5.6 times more often than you will win).
So that your hand deserves a turn call, you will have to extract much more on the river (36,5$ at least).
But there's even worse: not be ware that one of your opponents have a set and think you have the best hand when the river comes or .
Because, when you make your hand, not only will you lose the 15$ of the turn call but, you might be stacked (for 81$) on the river by a boat!
If you take that into account, you get the reverse implied odds of the pot.

As you see, calculating odds is not complicated mathematically. What is difficult is to estimate your opponent's hand, which will impact the number of your/his (supposed) outs.

If the flop reads and turn is still .
Then you are certain you have the best hand on the turn.
If you think one of your opponents has a set, then he has a redraw against you. That means that even if you make your hand on the turn, he still have a chance to draw to a better hand on the river (a boat).
A set would have 10 outs against your flush. From his point of view, the odds of his hands to win are 36/10=3.6 to 1.
So, if both your opponents check the scary turn, you have to bet a sufficent amount to make sure the pot won't lay them proper odds to outdraw you.
If you think both players might call (one has a set, the other is a fish), then bet the pot for instance (17.5$).
Both callers will have 3 to 1 pot odds on their call.
If both players fold, then you will make a net profit of 13.5$ on this hand.
If both of them call, one with a set, while the other has no out, and if you have the discipline to fold your flush when the set fills and bets on the river, then this hand will show a profit strictly superior to 13.5$ (because by your turn bet, you made your opponents make another mistake). The difference being this profit won't be a constant (like when they fold). Often, you will scoop a big pot and from time to time you will lose one but, on average, the profit will be greater than 13.5$.

Everytime an opponent calls a bet without sufficent pot odds, you increase your long run profit on this hand.
Everytime an opponent who has the best hand doesn't bet enough to cut your pot odds, then you reduce your long run loss on this hand by calling him.


A few probabilities to improve tour hand:
Extract from Howard Lederer's Secrets To Holdem Poker.

Texas Hold'Em
# of outs	type of draw	On flop	On turn
2	Pocket pair that needs to hit trips to win	9%	5%
3	Pair that needs to hit its kicker	13%	7%
4	Inside straight draw	18%	9%
5	Smaller pair on board with a live kicker	20%	11%
6	Two live cards to hit	24%	14%
8	Open ended straight draw	32%	18%
9	Flush draw	37%	21%
12	Flush draw with one live over card	45%	27%
15	Open ended straight and flush draw	54%	34%
21	Open endend straigth draw, flush draw, and two live other cards	67%	48%

A few examples:

1. Pair that needs to hit its kicker
It happens when you and your opponent share the same splitted pair but he has a better kicker or he has two pairs, his second pair being lower than your kicker.
You only have 3 outs, the three remaining cards of the same rank than your kicker. Indeed, if your pair becomes a set, you will still lose as your opponent will have the same set with a higher kicker or a boat.
Ex:
You=
Your opponent=
Flop=
Your hand is said to be dominated, you will have to be really lucky to win.

2. Smaller pair on board with a live kicker
It happens when you have a smaller splitted pair than your opponent but if you hit your kicker or if your pair becomes a set, you will win. Same thing if your opponent has two pairs but your kicker is higher than each of his cards.
Ex:
You=
Your opponent=

Flop=
This time you have 5 outs; 3 aces and 2 tens.

That's all for that first article. It might have looked a bit complicated to some but be sure that, with a bit of experience, it will become easy. You can comment this article on the forum.