Hi,
I must say that I had expected more from this article.
I'd like to address the mathematical part of it. The problem is that author presents some calculations without even trying to explain reasoning behind it. It seems that numbers are there just to justify his point. If there is no intention of explaining mathematical aspects then there really is no need to include them. It would be much better if author would write something like that:
The equation of calculating gross EV (if opponent calls our bet) is:
gross EV = p*(2*bet_size + pot_size) (1)
where p is probability of winning the hand based on opponents range of hands ...
and that should ideally be followed by explanation of equation (1). And then you could write by (1) we have:
Our gross EV at this point is: 0.7 * (2 * $2.25) + $2.75 = $5.9
and everyone would see that there is a typing error
There are some more typing errors of "same category"
Our net EV (profit) at this point is: $5.9 - $2.75 = + $3.15
His net profit expectation is: $1.77 - $2.75 = -$0.98
He “invests” $2.75
Bet size on turn was $2.25 not $2.75 which is size of the pot.
There is nothing wrong with getting net EV from gross EV but perhaps the more intuitive way would be to get net EV directly by formula:
Formula for calculating net EV (if opponent calls our bet) is:
net EV = p*(bet_size + pot_size) - (1 - p)*bet_size (2)
where p is probability of winning the hand and pot_size is size of the pot before our bet.
Notice, that one could easily obtain (1) from (2):
net EV = p*(bet_size + pot_size) - bet_size + p*bet_size
net EV + bet_size = p*(bet_size + pot_size) + p*bet_size
gross EV = p*(2*bet_size + pot_size)
which is exactly equation (1).
Let’s calculate Villains EV for this example:
His gross profit expectation is $5.9 * 03 = $1.77
His net profit expectation is: $1.77 - $2.75 = -$0.98
Calculating Villain's EV from Hero's point of view makes no sense. Villain could possibly consider - after Hero's turn bet - that Hero's range is something like: 33, 66, TT+, AT+. Maybe Villain is holding 66 and is way ahead of Hero's range, maybe he is holding A7 and is a huge underdog. Assume Villain is holding 5d4d. He's got 15 outs - thats about 32% chance of winning the hand. Instead of
His gross profit expectation is $5.9 * 03 = $1.77
(btw: why $5.9) you should perhaps say
If Villain holds 5d4d and he believes he'll win if he hits one of his outs (which is almost always a case - given Hero's range)
then his gross EV would be by (1):
gross EV = 0.32*(2*$2.25 + $2.75) = $2.32
and therefore his
net EV = $2.32 - $2.25 = $0.07
So Villain is making profit if he calls. What went wrong? Hero is offering Villain a 2,22 - 1 pot odds which is about 31%. Even without implied odds Villain is correct to made this call every single time. Of course Hero could prevent this by betting more. But then he would probably be called just by better hands so that clearly wouldn't be a value bet. All in all this hand is not a good example for value betting. Reasons are: Hero has a weak made hand and Villain's range is somewhat tight.
In the section on continuation bet, author writes:
As our opponent also folds hands like AQ and AK after our continuation bet, our move is profitable when we calculate the circumstances of the continuation bet.
Since we win the pot with a bet in 67% of the cases, we make a huge earning, even when we make a pot sized bet.
I probably misunderstood it but why would you risk a pot sized bet only to be called by better hands? If you know that Villain is going to call with strong hands (better than Hero's) but is always folding not made hands (even some better than Hero's hand) just bet minimum.
Another problem of the article is hand posting part of it. There really should be some information about type of the game, e.g. All hands in this article are from NL10, full ring, and both Hero and Villain have a 100BB stack size.
Basically all I wanted to say is, if there are some nontrivial mathematical calculations involved in article then the article should contain all of the necessary formulas for this calculations. Ideally formulas should be explained, so the reader would be capable of using them on his own. This really should be the case in articles - such as this - which serves learning purposes.