re: Poker & Odds
Originally Posted by CallmeFloppy
True, but I think it brings up a good point. If you don't hit your card on the turn, are you still going to play the hand to the river? If you can expect your opponent to bet the turn when you missed and you expect to fold, then you have to add that into you calculation for the hand.
I think too many people get stuck at looking at a hand being played out one way (though sometimes less than that) and then get lost when the board or opponent doesn't cooperate.


Pretty much whats in bold.
The rule of 4 is the probability of hitting on the turn
or river. The key word being 'or'. When a player is allin, we can use this rule since we are guaranteed to see both the turn and river.
However, if we calculate the probability of hitting on the turn or river when there is a chance of future betting, we are giving ourselves false a false probability to hit since it isn't guaranteed you'll see the river if you miss the turn.
Lets say you have a flush draw and have 9s outs. Let look at a few scenarios.
1.
The pot is $50 and your opponent bets $25 and is
all in. You have to call $25 to win a pot of $75.. If you do the rule 4, you have a ~35% chance to hit your hand or 1.86:1 odds of hitting.
We already know it's a +EV call since the pot odds > hand odds, but if we plug this into an EV formula, we get:
EV = [$won x .35]  [$lost x .65]
EV = [$75 x .35]  [$25 x .75]
EV = [$26.25]  [$16.25]
EV = $10
This means that, if you play this scenario out over a large sample size, you'll average $10 per hand.
2.
Now lets say the opponent isn't allin. You do the rule of 4, but miss the turn. On the turn, opponent bets out pot ($100) and you fold. Since you fold and didn't see the river, when we plug it into the EV formula, we have to make adjustments since you only saw the turn and not the turn+river. The odds of hitting your out on the turn is ~19%
EV = [$won x .19]  [$lost x .81]
EV = [$75 x .19]  [$25 x .81]
EV = [$14.25]  [$20.25]
EV = $6
This means that, if you play this scenario out over a large sample size, you'll average $6 per hand.
3.
Same as scenario 2, but this time you call the $100 bet to win a pot of $200. You have a ~20% to hit your out on the river.
If we plug this into the EV formula we get:
EV = [$200 x .20]  [$100 x .80]
EV = [$40]  [$80]
EV = $40
You're still losing money.
4.
Same as scenario 2, but villain bets out $25 on the turn. You have call $25 to win a pot of $125.
If we plug this into the EV formula, we get:
EV = [$125 x .20]  [$25 x .80]
EV = [$25]  [$20]
EV = $5
This looks like it's a +EV call, but remember you're still losing $6 on the turn. So, even if this is a +EV call, you're still $1 over the long run in for both situations.
The only way you would break even after making the EV call in scenario 2 is your opponent bet out $23 on the turn and you had to call a bet of $23 to win a pot of $123. When plugged into the EV formula we get an EV of +$6 + ($6) = 0. If they bet any less, then calling would be +EV.
Now, you might be wondering "What if, in the 2nd scenario, I call and hit the turn? Doesn't that mean anything?"
The answer is 'no'. Even if you hit your flush on the turn, we still calculate using the rule of 2 for the probability of hitting on floptoturn. Even if you hit the turn, the call is still EV and you're losing money in the long run.
In poker, it's about making +EV decisions. It doesn't matter if you hit your draw, if it's a EV call, you're losing money.
I hope this all makes sense.