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 Originally Posted by spoonitnow
I' m going to try to use an example to explain this with an example in the most straight-forward way I know possible.
Suppose you're on the river, heads-up, in position in a $10 pot with $10 remaining in the effective stacks. Your opponent checks to you with 15 combinations, 5 of which beat you. Suppose he folds x number of combinations if you bet, and that he never folds a hand that beats you.
The EV of checking is (5/15)(0) + (10/15)(10) = $6.67.
The EV of betting is (5/15)(-10) + (x/15)(10) + ((10-x)/15)(20). Let's quickly simplify the betting equation:
(5/15)(-10) + (x/15)(10) + ((10-x)/15)(20)
-3.33 + (2/3)x + 13.33 - (4/3)x
10 - (2/3)x
So when is the EV of checking greater than the EV of betting?
6.67 > 10 - (2/3)x
-3.33 > -(2/3)x
5 < x
Checking is better than betting whenever he folds more than 5 combinations. If he folds more than 5 combinations, that means he's calling with less than 10 combinations. Since 5 combinations beat us, if he's calling with less than 10 combinations, then our equity against his calling range is less than 50%.
Therefore, checking is better than betting when we have less than 50% equity against our opponent's calling range.
Hi guys, thanks for this very informative post. Just for the sake for answering my own question, I tried the following calculations:
Scenario 1: if we bet smaller on the river (Let’s go for the extreme, $1), wouldn’t we need to hv less equity against his range to be a better play than checking?
EV of checking remains the same as Spoonitnow’s calculation: 6.67
EV of betting: (5/15)(-1) + (x/15)(10) + ((10-x)/15)(11)
= (-5 + 10x + 110 – 11x)(1/15)
=(1/15)(105 – x)
For betting to be better than checking:
(1/15)(105 – x) > 6.67
105 – x > 100
5 > x
Hence villain has to fold less than 5 combos. Let’s say he folds 4. Hence he’s calling with 11 combos, out of which 5 beats us. Hence we have 6/11 equity against his rang. To conclude, our equity against his calling range has to be greater than 50% for our bet to be better than our check. Hence, whatever amount u bet on the river, u still need to be stronger than his calling range 50% of the time to make it a better EV play than checking! (Duh… Why would I think otherwise in the first place…..?)
This is where I had another question. The above scenario is based on the assumption that he’ll never fold a hand that is better than our range. What if we bet the pot and he folds, let’s say, 1 or 2 of his 5 combo that beats us?
Scenario 2, if we bet pot: $10, and he folds 1 combo that beats us:
EV of checking remains the same: 6.67
(Assume he folds y hands out of the 5 that beats us)
EV of betting = ((5 – y)/15)(-10) + ((10 + y)/15)(10)
=(2/3)(5 + 2y)
For EV of betting to be great than EV of checking:
(2/3)(5 + 2y) > 6.67
5 + 2y > 10
y > 1/2
Hence, for our bet on the river to work as a semi-bluff, we have to make him fold more than half of his hands which are beating us.
So basically now we’re turning the hand into a bluff. I've done calculations assuming opponent's range beating us 10/15 of the times, and the results are the same. We still hv to make him fold 50% to make the bluff profitable.
So... I guess the conclusion here is... u hv to make villian folds more than half of his beating us range to make ur bluff on the river to be profitable.
Someone please discuss about this.
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