I just came back from running. The track was closed so I had to jump the fence but it was very relaxing: I was the only one running, the lights were off so everything was dark and you could see the stars. It was a little bit cold but I ran with my North Face jacket on so hopefully I won't get sick tomorrow...
I've been thinking about a very basic concept in poker which is usually misunderstood by beginners: the number of outs available to make your hand and how to calculate your odds. "Outs" are those cards that can complete your draw and give you a winning hand. It is important to keep in mind that you rarely know how many outs you really have because your opponent's hand is unknown. However, if you're able to put your opponent on a reasonable range of hands you can usually figure out how many outs you have and whether it is profitable to call your opponent's bet in order to "buy" an extra card and try to make your hand.
Here is an example of how sometimes a drawing hand can actually be ahead of a made hand due to the enormous amount of outs left in the deck:
No-Limit Hold'em, $1.00 BB (6 handed)
UTG ($66.12)MP ($102.15)CO ($111.55)
Button ($119.78)SB ($135.55)
Hero (BB) ($100)Preflop: Hero is BB with 8

, K

UTG calls $1,
MP raises to $2,
1 fold, Button calls $2,
1 fold, Hero calls $1, UTG calls $1
Here MP makes a very small raise and he gets 4 callers (I'm in the big blind and I call the extra dollar with a weak hand because there are so many callers that I'm getting good odds to call with pretty much any random hand)
Flop: ($8.50) 9

, J

, 10
(4 players)Hero checks,
UTG bets $8.50,
MP raises to $34,
1 fold, Hero calls $34,
1 foldMonster drawing flop for me: I get an open ended straight draw and a flush draw. I'm assuming any club, any 7 or any Q will give me a winning hand so I have 9 outs for the clubs, 3 outs for the three sevens left in the deck (the 7c has already been counted as part of the 9 clubs) and 3 Queens (Qc has also been included in the 9 clubs that will give me a flush). Therefore, I have 15 outs.
However, the action before me has been very strong: UTG has bet pot and MP has re-raised pot. This strong action leads me to believe that MP has a made hand that he is trying to protect from a club draw. Therefore, he probably doesn't have any clubs and my outs to the flush draw are good. He could have a made straight with QK, an overpair (QQ, KK, AA) or a set (99, TT, JJ).
At this point there are $8.50 + $8.50 + $34 = $51 in the pot and I need to pay $34 to see an extra card. Is it a profitable call?
Options:
- Opponent has an overpair (QQ, KK, AA). In this case it's a coin flip: 50.5% for my opponent vs 49.5% probability of winning the hand for me. Therefore, in this case my call is clearly profitable: Net cost to play = -$34; Pot value = $17 + $34 bet + $34 call; Odds to win = 49.5%; Net Profit = Net cost to play + (Odds to win * Pot Value) = - $34 + (0.5 * $85) = $8.5 per hand. Note that these are only the exact pot odds but my opponent still has a $66.15 stack behind him so my implied odds are much better (Implied Value is a cool concept that takes into account future betting but we'll leave that for another day).
- Opponent has a made straight with KQ or 78. In this case he is a 56% favorite vs my hand. Therefore, my EV is: Net cost to play = -$34; Pot value = $17 + $34 bet + $34 call; Odds to win = 44%; Net Profit = Net cost to play + (Odds to win * Pot Value) = - $34 + (0.44 * $85) = $3.4 per hand. Still a profitable call even if my opponent already has the straight!
- Opponent has a set (99, TT, JJ). In this case I'm in worse shape: I'm a 61%-39% underdog so my EV is: Net cost to play = -$34; Pot value = $17 + $34 bet + $34 call; Odds to win = 44%; Net Profit = Net cost to play + (Odds to win * Pot Value) = - $34 + (0.39 * $85) = -$0.85 per hand. I would be losing $0.85 per hand in the long run if my opponent always had a set in this situation.
Now, I really don't know what my opponent has but I could say he has a 33% chance of having each of the hands discussed above. Therefore, my combined EV would be 0.33*$8.5 + 0.33*$3.4 - 0.33*$0.85 = $3.65 per hand on the long run.
Note that the expected value calculated above does not take into account the fact that my opponent still has $66.15 left. The odds of making my hand are computed based on two cards left to come (turn and river) and my opponent may go all-in on the turn so I may have to pay an extra $66.15 in order to get to the river. At this point, however, I'm pretty much decided to go all the way.
Turn: ($85) 6
(2 players)
Hero checks, MP bets $66.15 (All-In), Hero calls $64 (All-In)
The 6s is not one of my outs and my opponent goes All-in. At this point the expectation of calling is still positive despite what some people might think:
- Overpair (AA,KK,QQ): Hand odds are 60%-40% for my opponent. EV = Net cost to play + (Odds to win * Pot Value) = -$64 + (0.40*$213) = $21.2
- Straight (78, KQ): Hand odds are 74%-26% for my opponent. EV = -$64 + (0.26*$213) = -$8.62
- Set (99,TT,JJ): Hand odds are 70% - 30% for my opponent. EV = -$64 + (0.30*$213) = -$0.10
Therefore, the combined EV is given by: 0.33*$21.2 - 0.33*$8.62 - 0.33*$0.1 = $4.12
River: ($213) 7
(2 players, 2 all-in)
River is one of my outs and I make my straight
Total pot: $213 Rake: $3
Hero had 8
, K
(straight, Jack high).
MP had K
, K
(one pair, Kings).
Outcome: Hero won $210
My opponent had an overpair (the worst hand he could have) and I take his money... "iiiiiiiiii" <-(I'm happy when I win), he should have made a much larger raise preflop in order to eliminate as many players as possible from the hand.
My opponent's comment at the end of the hand was "you fucking lucky donkey"... These kinds of comments amuse me endlessly for two reasons: first of all, it's always entertaining to make other poker players mad and, more importantly, it shows that people don't have a clue of how to play: I made the correct play on the turn and on the river. As I've shown above, this play has a positive expectation, that is, if I repeat the same play a sufficiently large number of times I will make money. You could argue that the 7h on the river has a lucky card in this particular situation but, at the end of the day, winning or losing at poker has little to do with luck.