Having a problem figuring out values of actions on a hand.

[Daliman]

Let's say I am in a hand where I am in position on the river with an 85% chance I have the best hand. When I bet and win, I make $10, When I bet and am losing, I lose $20 though, as I am check-raised. It's pretty easy to figure out what the value is of betting this every time;

.85 x $20 = $17
.15 x $40 = $6
17 - 6 = $11

So each bet is worth $11 if bet 100% of the time, (right?).

But what if I bet 85% of the time randomly? Or 90%? Or 75% How do I figure for these. I made a few attempts at it, and the #'s I come up with just don't look right at all. I'm trying to figure mainly for this problem at 75, 85, 90, and 95%. ANy help would be greatly appreciated.

[chief444]

85% of the time you win 10.
15% of the time you lose 20.

.85*10 - .15*20 = 5.5

If you randomly bet then you randomly win an average of 5.5.

Realistically you'll win some of the time you're check/raised or you shouldn't be calling.

[Daliman]

As an after-the-fact edit to my previous post, the figures were for a 10-20 game, but I posted it as using a 5-10 game. My apologies on any confusion.

[Daliman]

[ QUOTE ]
85% of the time you win 10.
15% of the time you lose 20.

.85*10 - .15*20 = 5.5

If you randomly bet then you randomly win an average of 5.5.

Realistically you'll win some of the time you're check/raised or you shouldn't be calling.

[/ QUOTE ]

Other than my initial posting error, you are initially saying exactly what I said as a value, but this is more a game theory question in the fact that if you win 85% of the time, shouldn't there be a breaking point where either betting more than a certain % gives you no extra gain or little gain, and possibly even a loss?

[ QUOTE ]
Realistically you'll win some of the time you're check/raised or you shouldn't be calling.

[/ QUOTE ]

I left this out of the equation as ancillary at best, because also, some of the times you bet you will not be called, among other things, which more than offsets the CR wins.

[chief444]

[ QUOTE ]
Other than my initial posting error, you are initially saying exactly what I said as a value, but this is more a game theory question in the fact that if you win 85% of the time, shouldn't there be a breaking point where either betting more than a certain % gives you no extra gain or little gain, and possibly even a loss?

[/ QUOTE ]
Sorry Daliman...didn't mean to nitpick I just thought it was actually a mistake.

As far as the game theory question, the only way you shouldn't bet if you're ahead 85% of the time is if he won't always call. So with the assumptions listed any time you bet you make the $11 on average and any time you check you make $0. If you say the opponent only calls with the top xx% of his hands but folds or check/raises the rest then there will be an xx% where it becomes correct for you to just check. I guess what I'm saying is it depends on your opponent and we need to make some more assumptions to get into anything like you're asking right?

[WhiteWolf]

Giving the parameters of your post, not betting has an EV of 0. Betting with a frequency of x, the equation for overall EV becomes:

(.85 * x * 20) + (.15 * x * -40) + [(1 - x) * 0]

Simplifed:
x * [(.85 * 20) + (.15 * -40)]

Ultimately,
x * 11

So betting with frequency 85% yields an EV of $9.35 per hand.

The parameters given in your EP make it clear that betting 100% of the time is the optimal EV strategy.