Mathematics of NLHE Ep 4

Covered in this episode:

• EV calculations
• EV calculations
• and more EV calculations

Expected Value (EV) is the long term expected outcome of a given hand or situation, either positive (+EV), negative (-EV), or neutral (0EV).

EV Calculations 101

Basic EV calculation setup:

• EV = [result of win] - [result of loss]
• To expand a bit:
• EV = [Our Equity] * [what we win] - [Villain's Equity] * [what we lose]

• Basic Example:
• We're in the big blind w/ AsAc. The UTG player shoves w/ KhKd and folds to you. It's $990 to call to win $1015 (stack + blinds) 5/10nl, $1000 stacks.
• EV = (.81 * 1015) - (.19*990)
• EV = 822.15 - 188.1 = +$634.05

• Alternative Method:
• EV = [our equity] * [total pot] - cost of our call
• EV = (.81 * 2005) - 990 = +$634.05

Example 1 - 89o on 7T24 vs AA. We need to call 650 to win 700, what is our EV for calling?

• EV = .18 * [650+700] - .82 * 650
• EV = 243 - 533 = -$290

Example 2 - Vil opens from the to Co to 35, we 3bet him again OTB to 125, Vil thinks for about 2 seconds and calls.

• First let's put him on a range: Mostly, pairs, AQ (he would 4bet AK a lot), occasionally a suited connector or AXs type hands.
• On the flop Kc 9c 8d, Vil checks, we bet $200 into $265, he checkraises all in for $875 meaning it costs us another $675 to call. Do we call or fold?
• Let's evaluate his range now that we have more information: He always has 8 outs+ if drawing (OESD, FD or better), and let's assume he 4bets AK 100%, but he could call AA/KK planning to trap.
• Based on our preflop and flop range, we now get a narrowed down range of: KK+,99,88,AQcc,AJcc,ATcc,A8cc,QJcc,QTcc,JTcc,98s,87cc,JTs,67s. Against this range, how is our hand doing?
• Against the range above, our hand has about 35% equity

• EV(call) = .35 *(875+200+265) - .65 * 675
• EV(call) = 469 - 438.75 = +$30.25

WoT's TUPAC method

• Following these 4 steps will help guide you to mentally calculate your equity against a hand range while at the table:
1. Tally Up the hand combinations
2. Pair combos to known equities
3. Analyze unpaired combos
4. Combine the analysis to estimate

Cont. on Example 2 using TUPAC method

1. Tally Up the hand combinations
• Let's break apart his range into the 3 main categories of hands: Those that crush us, those we're flipping with, and those we're decent favorite over.
• Crush us: AA, KK, 99, 88, 98s (5 hands, 13 combos)
• Flipping with us: AQcc, AJcc, ATcc, A8cc, QJcc, QTcc, JTcc, 87cc, 67cc (9 hands, 9 combos)
• Decent favorites against: 67s, JTs (2 hands, 6 combos)

2. Pair combos to known equities
• Start matching up hands that crush us and the flips, those should even out between 25 to 35% equity depending on how badly we're crushed. A set has us drawing much thinner than overpairs.
• There are 13 combos of hands that crush us and 9 combos of hands that flip. If we match those, we will have 9 pairs with about 25-30% equity (not quite because of the sets) and 4 left over unpaired combos that crush us. We can guesstimate that we have about 30% equity before the 4 unpaired combos and after adding in the left over combos, our equity is going go down and be between 25 to 30%

3. Analyze unpaired combos
• There are 5 combos of OESDs and against those hands, we are a 2:1 favorite or 66%
• Thinking a little deeper though, we have a Q and that removes one out from the JTs hands which moves us closer to a 3:1 favorite.
• Our JTs can average out with the left over combos of 98s from the "crush us" category which will boost our between 25 to 30% lets say 27% a percent or 2 to about 28% or so.
• That leaves us with the 3 unaccounted combos of 76s where we are about a 2:1 favorite. 76s makes up about: 3/(13+9+5) or 10.7% and we have 66% equity so .66*.107 = 7%.

4. Combine the analysis to estimate:
• Our base is about 27% after pairing the combos of crush and flips
• Pairing JTs and 98s bumps our equity up a little
• Our decent favorites add about 7%
• The total is 27+1+7 = 35%
• This is very close to the 34% equity that pokerstove gives us. This method is by no means 100% accurate and at times, it will be incorrect but this is a good way to start calculating equities and as you get better with pokerstove and estimating equities, this will all become easier and second nature to you.
• Finally when our decision at the table is so close, where our estimated equity calculation is close but not 100% in line with our mathematical requirements, it is best to rely on the gut reads/intuition to tip the scales on whether to call or fold, depending on how frustrated we perceive our opponent to be.

Alternate Method

1. Tally up combinations
• Crush us: 13 combos w/ about 13% equity
• Flipping: 9 combos w/ about 50% equity
• Decent favorites: 6 combos w/ about 66% equity

2. Multiply range by equity
• (13/28) * .13 = .06
• (9/28) * .5 = .161
• (6/28) * .66 = .141
• You can fudge these numbers to make it easier
• Ex. (13/28) is close to 50% so .5 * .13 = .075. (9/28) is close to 30% so .3 * .5 = .15. (6/28) is close to 20% so .2 * .66 = .132

3. Add up the equities
• .06+.161+.141 = 36%
• Fudge: .075+.15+.132 = 36%

Table Estimations Thought Process

• WoT's thought process at the table when using his TUPAC method:
• As soon as I get checked raised all in, I think: ok, I doubt he ever does this as a pure bluff and right there I am talking about his hand range. I figure him to do this with big draws, sets, pocket aces, and the two OESD, 67s and JTs. He never has AK here because he will 4bet it 100% preflop. There are about 10 or so big draws he can have depending on how many Axcc combos and suited club broadways he is calling with preflop to a 3bet. I know I have been hammering on him so I know he may call with a lot of those combinations. So the more he is calling with those combinations the more draw combos there will be. As for sets, it's hard for him to have KK because I have one and the board has one. He could have AA and the smaller sets. He has aces and smaller sets about as often as he has big draws so if that's true, my equity against that part of his range is between 25-30%. he could have the 98s for two pair and the OESD as well. I am doing bad against the two pair but I am ahead of his OESD so that sort of balances out and gives me some equity. I am actually a 3:1 favorite against JTs since I have one of the Qs removing an out for him, giving me more equity. Over all I'd estimate that I am about 30-35% or about a 2:1 dog. I am getting about 2:1 on my money which makes it close. This leaves it down to reading my opponent. The more frustrated he is, the more apt I am to call. The less frustrated he is, the more I will fold.