Poker odds

Anyone fancy explaining how to work out the probabilities of cards coming up. Was never any good at probability in school and I'm still not now (when I actually need it!).
If someone could just explain how it works when you're looking for a card before the flop, turn etc.
It would be much appreciated. The more examples the better. Cheers.

The probability is number-of-cards-that-will-help-you divided by number-of-unseen-cards.

For example, four to a flush in hold'em on the flop has a 9/47 = 19% chance of improving to a flush on the turn, because there are 9 cards that will give you the flush, and there are 47 cards that you haven't seen.

Note that how many cards that have been dealt is of no consequence.

I'll go into more detail if you need it, but let's start there.

A very rough guide to working out probability of hitting your hand after the flop is work out the possible outs (the cards that will make your hand) and times it by 4 if you've got 2 cards to come, and by 2 if you've got one card to come (so by using F Paulsson's example above, 9 outs with one card to come = 18% of hitting flush, so it's only 1% out). You then work out what return you are getting on your bet if you call - if the return is better than the odds of getting your hand then you call, if the returns are worse than the odds you fold. You dont go on flush draws against one opponent, but you can against 2 or preferably 3.

What's the best way to know if the return is better or worse? What do I base this on? What if I'm in on a draw with AK and already have a nice investment against just 1 player who probably has high or 2 pair?

Typically, one talks about "pot odds", and there's also "implicit odds".

Pot odds:
The idea is that your bet should be profitable in the long run. So if the pot is 100, and you need to call a bet of 20, and the next card will mean you win, then you're getting "pot odds" of 100-to-20, or 5-to-1.

So if your chance of getting the winning card is GREATER than 5-1, then you're getting "good odds" on that draw and will win money in the long run. In the case of a flush draw, you're getting about 4-1 (81-19). If you pay 20 into a pot of 100, and you repeat this 100 times, you'll end up like this:

81 times, you'll miss the flush. Cost: -20*81 = -1620
19 times, you'll hit the flush and win. Profit: 120*19 = 2280.

2280 - 1620 = 660. Your EV (Expected Value) of that call is, in other words, $6.60 per hand.

Implicit odds are different, but let's start here. Let me know if that makes sense or if you want me to explain further (or if you were asking something completely different altogether).

That makes it perfectly clear to me. Thank you FP. No wonder you hit the rep. ladder so hard and fast.

I think I could eventually learn this (I'm bad at math, especially in my head)) but to do it in the time they give you (online) I don't think I could ever come up with the right answer. I think this is hurting my game tremendously and I need to work on this aspect if I'm ever to improve.

I use a variation of Rob's way of counting my odds:

n*2 +2

... where "n" is the number of outs I have. So if I have a flush draw, I have 9 outs (as discussed earlier). This gives me: 9*2+2 = 20%.

Adding a few more options in there, let's say you have:
A♥K♥

And the board is
10♥J♥2♣

If you think your opponent has a pair of a jacks (but not a king or an ace kicker), you now have the following outs:
Any ace = 3
Any king = 3
Any queen = 4
Any heart that's not a queen = 8.

... = 18 outs. The simplification above gives 18*2 + 2 = 36+2 = 38. So now, you only need odds of 5-2 or better in order to call (because 100 to 38 is about 10-4 is about 5-2).

So if the pot is $10, and you need to pay another $2, you're getting much better odds than your draw requires. If you had to pay $5, however, you'd be getting bad pot odds - but quite possibly good implicit odds. But more on that later!

Having some time to kill early in the morning at work, I figured I'd move on to implicit odds, if the same readers want to know more about that, as well:

Where pot odds take into consideration the money that's in the pot right now, implicit odds is an estimation on how much money you CAN win from the bet if you hit one of your outs. For instance, with 100 in the pot, and a bet of 20, is your gain really only 100 if you win? Can you really not squeeze out an extra few bucks from your opponent if you hit your flush? You probably can - and so as the pot will get bigger, your implicit odds go up.

A good example of when implicit odds come into play is when you limp in with a small or medium pair before the flop in hold 'em. Your chance of hitting a set (which is typically the only way a small or medium pair will win) is around 7.5-1, which means that pot needs to have 6 or 7 other limpers to make it worthwhile. But, of course, that's presuming that everyone will fold if you hit your set, which is rarely the case. Let's say instead that you get four other limpers and will narrow the field down by 50% on the flop, and another 50% on the turn - what are your implicit odds?

Four limpers to the flop = 4 SB.
Two callers to the turn = 2 SB.
One caller to the river = 1BB = 2 SB.

Here, you stand to win 8 small bets, at the initial price of 1, which gives your call positive expectation. By this count, your implicit odds are good to make this pre-flop call with a weak pair because of the money you'll figure to win if you do hit your set, rather than the amount you're "guaranteed" to win.

Here's the downside to implicit odds though: They're an estimation, and as it so happens, people tend to be way too optimistic in calculating them. For instance,

K♥7♦

on a board with
Q♥9♣8♥A♥

gives you 9 outs to a flush, which is a 4-1 shot. Now let's say that there were only two of you in the pot, one limper and you in the BB. Flop was checked around, and he bet at the turn after you checked, the pot would be about 2BB. You're paying 2-1 to see the last card, which could give you the nut flush - but do you call? Pot odds say no. Implicit odds likely don't give you the numbers you're looking for either, but this is where people get overoptimistic!

If your opponent paired his ace and has no hearts, would he really bet into a four-suited board after the river? Would he call your bet? Probably not. You can hardly figure to win more than the money that's already in the pot at the turn, because if you make your hand on the river, he's not going to pay you enough. Even if he calls an extra bet on the river (maybe he has J♥), you're still not getting good enough odds. At that point, your call on the turn will have cost you 1BB, and you're looking at a profit of 3BB, which gives you 3-1. You'll have to successfully checkraise him (and he has to call your checkraise) for it to be near profitable, and you have to succeed at that every time that you hit your flush. Hardly likely.

I know some players who think this is mathematical mumbo-jumbo and has no place in a gambler's heart, but this is really the principle that separates winning players from losing players: Being able to tell a profitable bet from a non-profitable one. In the example above, there's a non-profitable bet being offered. Don't take it.

Cheers,
FP