[DavidC]

[ QUOTE ]
If, otoh, you think there's greater than a 50% chance you're not dead, you raise.

[/ QUOTE ]

This is something that I'm interested in learning, because it's common sense that you're right about this, but you're wrong. I'm pretty sure that you know this, but I'll go over it just so we don't miss anything when putting together our formula. Actually, you'd be pretty damn close to correct if hero had 2bb or less remaining, or if villain only had 1bb left in his stack.

The formula for raising the river heads-up is easier than the one for determining if you should bet the river out of position or not. Could you be so kind as to make the formula for that one, since I did this one? I'd be happy to nit the hell out of it when you're done, so that we can get as close to a true answer as possible...

--

Let:
A) chance that you're ahead and he calls
B) chance that you're ahead and he raises and you call
C) chance that you're behind and he raises and you call
D) chance that he's ahead and he calls your raise
E) chance that you're ahead and he folds
F) chance that you're behind and he folds
G) chance that you both have the same hand and he folds.

For the purposes of this discussion, I'm going to say that F and G never happen. If it does, though, you have to understand that the factor by which you multiply F is the pot size, and G is half the pot size, whereas other factors are multiplied as shown below. Therefore in large pots, when F and G exist, they have a very large impact on the EV even at low values of F and G. Since E has factor of zero, it's removed from the formula, but it's taken into consideration when creating the values of A through D.

Also, even though the NLHE equivalent of this formula is The One True Formula (which can be applied to limit), it would make my brain hurt, so we're assuming both opponents are playing with infinite stacks in a LHE game.

EV = 1A+2B-2C-1D*
* please note that I'm using 2B and 2C because of the instant profit/loss of 1bb occuring no matter what

Trial with your C value shows:
Let: A=48%, B=1%, C=49%, D=1%, E=1%
= 1(0.48) + 2(0.01) - 2(0.49) - 1(0.01)
= 0.48 + 0.02 - 0.98 - 0.01
= -0.49BB as a result of raising the river when C = 50%.

Therefore if there's a 50% chance that your opponent is ahead, you should definitely not be betting, according to this formula.

Edit: in this scenario, if villain three-bets us, we should fold unless the pot is greater than 99 bets, so we may need to tweak the formula under the assumption that we fold if he three-bets. Or we need to tweak the values to make a high enough value for him three-betting with a beaten hand that it makes sense to call him on the river, or we have to accept that we're just not good enough to fold the river when three-bet, even though we know we're not getting good enough odds to call... regardless, we have to tweak the numbers a bit to determine where the break-even point is, in order to achieve expert river play.