I was reading a thread in the Blind Bet Poker Forum where a player
described a live cash game situation where he lost a whole buy-in and
wondered about the odds and probabilities of the situation. I thought this
would be a good opportunity to detail a probability calculation in an
article.
Do these types of calculations offline, when you’re out of the game.
Making the time then to understand these situations fully will make you a
much better poker player when the heat is on!
The Hand
The player held 4 2 in the big blind. Amazingly, every single player at
the table limped in. Our hero checked his option (which is probably wise
with 4 2 and eight opponents); so nine people saw the flop.
I probably don’t need to tell you that the flop contained three clubs. Our
hero flopped a flush, albeit not a very good one with only a four-high.
Yes, I know it’s impossible to have a “four-high” flush, but I mean to
have a favorable comparison against another player who might be holding,
for example, 7 6. That player would have a seven-high flush in my
convention, even if the board cards were higher.
After lots of betting, our hero ended up all-in on the turn with three
opponents, one of whom held the A, another who held the K, and another
who held the Q 7. Our forum member asked,
“Think about the odds for a minute.... I have a 0.84 percent of flopping a
flush on the flop and I don't even know the odds that two people flop a
flush and two more flop flush draws so what I want to know is could I have
put someone else on a flush already??? I put them on draws but could I
really put someone on a flush?”
My intention here is not to analyze the betting of the hand and find
reasons for different bet sizes at different points. I will make a comment
on the game, however. A live no-limit cash game will play much looser than
a comparable limit game in online poker. Seeing 100% of the players taking
a flop is amazing, but very risky. When you play a game like that,
especially in no-limit, you are essentially playing primarily implied odds
at every street. “Betting on the come,” meaning to bet when on a draw, is
the normal procedure here. Don’t be surprised if a flopped set gets beat
by a drawn-out flush in this type of game.
The Odds
Now I’ll go back to the player’s question about the odds. I will analyze
them here. Our hero held two clubs and three were on the flop. The
question is, what are the odds of at least one other opponent having
flopped a higher flush?
First of all, any other opponent holding two clubs will have a higher
flush, since even a 5 3 beats
our player’s 4
2. So there are eight
clubs out (our player has two and the board has three). There are 47
unknown cards. What are the odds that one other player has any two clubs?
To answer this question, take the five known clubs out of the deck and
deal two to each of the other eight players. We go ahead and give our hero
the 4 2 and put the other three down on the table. Since the deck is
randomized by the shuffle, it doesn’t matter what order we deal the
opponents’ hole cards in, so we will give each player two cards
consecutively.
There are four possibilities for each player:
- Both cards are clubs
- The first card is a club and the second is not
- The first card is not a club and the second is
- Both cards are not clubs
I will illustrate the odds calculations for player one.
Player One: the first card is dealt, and there is an 8/47 chance of this
being a club. If it is, there is a further 7/46 chance that the second
card is also a club. Multiplying the probabilities together gives us an
(8/47) * (7/46) = 2.6% chance of player one receiving two clubs.
There is an 8/47 chance that the first card is a club, followed by a 39/46
chance that the second is not, for a possibility of 14.4%.
There is a 39/47 chance that the first card is not a club, followed by an
8/46 chance that the second is a club. This is also 14.4%.
Finally, there is a 39/47 chance followed by a 38/46 chance that both
cards are not clubs, for a net probability of 68.5%.
After each player receives two cards, we will make a determination. If
that player received two clubs, we will assume they will call all-in,
meaning our hero is drawing dead. Our calculation would then end, because
all we care about is the likelihood of being beaten on the flop. We are
not considering that the bare A might call and draw to a better flush.
The probabilities change after each opponent, depending on whether they
got one or zero clubs. If Player One got one club (28.9%), the
calculations for Player Two go as follows:
- Chance of two clubs (7/45) * (6/44) = 2.1%.
- Chance of one club (7/45) * (38/44) = 26.9%.
- Chance of zero clubs (38/45) * (37/44) = 71%.
If Player One got zero clubs (68.5%), Player Two’s chances are as follows:
- Chance of two clubs (8/45) * (7/44) = 2.8%.
- Chance of one club (8/45) * (37/44) = 14.9%.
- Chance of zero clubs (37/45) * (36/44) = 67.3%.
Now, I won’t bog this article down with an exact detail of each
possibility. Besides, there are much simpler ways to perform these
calculations using statistical tools. Any poker player who has designs on
becoming a professional should definitely consider learning basic
statistics to help simplify this type of analysis.
The Answer
The end result, after we deal two cards to each of the eight players
remaining, is the probability that at least one of the eight players will
receive two clubs is 0.7694. Bottom line? Our hero had a 77% chance of
being a dead duck on the flop, and that’s without considering the
possibilities of a player with the bare A calling and filling the flush,
or of a flopped set drawing to a full house. Without doing those
calculations, I’d say it’s reasonable to suspect our player was more than
a 4-1 underdog on the flop.
The analysis? This hand might be worth a small bet due to the scare cards
of the flush, but if you are raised or called, you absolutely must be done
with this hand. Next time, it’ll be a no-brainer for you. You learned
another thing here, too: playing small suited connectors for flush value
is a major leak. Start folding 4 3.
You’ll be glad.
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