HM Quickie 1. EV Analysis Without Explicitly Assuming a Villain Range
Here is my first HM Quickie. A Hold’em Mathology Quickie will be some poker math issue that came up recently, usually through my participation in poker forums, that I thought would be useful to post but that doesn’t require the time, effort or detail for a full posting. It will generally be short and require that the reader have an understanding of basic poker math.
In a poker forum a poster wanted to confirm his EV analysis. He held AThh with a flop of 8h 3h Kc. With a pot of 380, villain bet 370, all-in. Hero could call or fold, the latter having an EV of 0.
EVcall = eq*(Pot + Bet) – (1 – eq)*Bet
Hero has the nut flush draw which has 9 outs. The 2x 4x rule then would provide an estimate of 4*9 = 36% for a hit. The exact hit percentage is
Pr(hit on turn) + P(don’t hit on turn but hit on river) = 9/47 + 38/47 * 9/46 = 35%.
Assuming that a flush hit wins the hand, 35% would be hero’s equity and his EVcall is as follows:
EVcall = 0.35* (380 + 370) – 0.65*(370) = 22
So, a positive EV would suggest a call is profitable compared to a fold.
The poster also asked what is the maximum bet size for a +EV call. If we let X represent the bet amount and set EVcall to 0, we have
EVcall = 0.35*(380 + X) – 0.65*X = 0
After a little algebra,
X= 380 * 0.65/0.35 = 443.3,
assuming both player’s stack sizes have that amount.
A comment by a very competent poster followed:
“It's impossible to calculate the largest bet size you can call (and at least break even) unless you provide villain's exact range.”
While I understand the thinking, I don’t agree with the “impossible” conclusion. As an example, if you know how many outs you have for the absolute nuts, you then know your card equity and can employ the applicable EV equation to solve for other variables.
Also, you never or at least rarely know villain’s exact range. I stated in the analysis I did that I assumed that given the nut flush hits, it’s a win. That implicitly defines opponent’s range. The best villain can have is KK, 88 or 33 each of which has 7 outs for a full house or quads, thus beating a flush. But the chance for those villain hands is only 3 * 3/C(47,2) = 0.0083. And, as noted by the commenter, if a limited range is considered, the probability of KK, 88, or K8 may rise to about 4%. Also noted, hero could win without the flush with his AT, so perhaps the win/lose effects negate each other. There are other possible situations such as hero hitting a straight or villain having two pair. Again, it is not unreasonable to assume a balancing out if there is no obvious advantage to one side.
Certainly, doing an analysis based on ranges is preferable but I do not think that not having a range estimate makes an EV-related estimate impossible if factors not explicitly considered are accounted for by reasonable assumptions such as very small occurrence probabilities or a balancing of plus and minus factors.