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The main underpinning of poker is math – it is essential. For every decision you make,
while factors such as 3️⃣ psychology have a part to play, math is the key element.
In this
lesson we’re going to give an overview of 3️⃣ probability and how it relates to poker. This
will include the probability of being dealt certain hands and how often 3️⃣ they’re likely
to win. We’ll also cover how to calculating your odds and outs, in addition to
introducing you to 3️⃣ the concept of pot odds. And finally we’ll take a look at how an
understanding of the math will help 3️⃣ you to remain emotional stable at the poker table
and why you should focus on decisions, not results.
What is Probability?
Probability 3️⃣ is
the branch of mathematics that deals with the likelihood that one outcome or another
will occur. For instance, a 3️⃣ coin flip has two possible outcomes: heads or tails. The
probability that a flipped coin will land heads is 50% 3️⃣ (one outcome out of the two);
the same goes for tails.
Probability and Cards
When dealing with a deck of cards the
3️⃣ number of possible outcomes is clearly much greater than the coin example. Each poker
deck has fifty-two cards, each designated 3️⃣ by one of four suits (clubs, diamonds, hearts
and spades) and one of thirteen ranks (the numbers two through ten, 3️⃣ Jack, Queen, King,
and Ace). Therefore, the odds of getting any Ace as your first card are 1 in 13 3️⃣ (7.7%),
while the odds of getting any spade as your first card are 1 in 4 (25%).
Unlike coins,
cards are 3️⃣ said to have “memory”: every card dealt changes the makeup of the deck. For
example, if you receive an Ace 3️⃣ as your first card, only three other Aces are left among
the remaining fifty-one cards. Therefore, the odds of receiving 3️⃣ another Ace are 3 in 51
(5.9%), much less than the odds were before you received the first Ace.
Pre-flop
Probabilities: 3️⃣ Pocket Pairs
In order to find the odds of getting dealt a pair of Aces,
we multiply the probabilities of receiving 3️⃣ each card:
(4/52) x (3/51) = (12/2652) =
(1/221) ≈ 0.45%.
To put this in perspective, if you’re playing poker at your 3️⃣ local
casino and are dealt 30 hands per hour, you can expect to receive pocket Aces an
average of once 3️⃣ every 7.5 hours.
The odds of receiving any of the thirteen possible
pocket pairs (twos up to Aces) is:
(13/221) = (1/17) 3️⃣ ≈ 5.9%.
In contrast, you can
expect to receive any pocket pair once every 35 minutes on average.
Pre-Flop
Probabilities: Hand vs. 3️⃣ Hand
Players don’t play poker in a vacuum; each player’s hand
must measure up against his opponent’s, especially if a player 3️⃣ goes all-in before the
flop.
Here are some sample probabilities for most pre-flop situations:
Post-Flop
Probabilities: Improving Your Hand
Now let’s look at 3️⃣ the chances of certain events
occurring when playing certain starting hands. The following table lists some
interesting and valuable hold’em 3️⃣ math:
Many beginners to poker overvalue certain
starting hands, such as suited cards. As you can see, suited cards don’t make 3️⃣ flushes
very often. Likewise, pairs only make a set on the flop 12% of the time, which is why
small 3️⃣ pairs are not always profitable.
PDF Chart
We have created a poker math and
probability PDF chart (link opens in a new 3️⃣ window) which lists a variety of
probabilities and odds for many of the common events in Texas hold ’em. This 3️⃣ chart
includes the two tables above in addition to various starting hand probabilities and
common pre-flop match-ups. You’ll need to 3️⃣ have Adobe Acrobat installed to be able to
view the chart, but this is freely installed on most computers by 3️⃣ default. We recommend
you print the chart and use it as a source of reference.
Odds and Outs
If you do see 3️⃣ a
flop, you will also need to know what the odds are of either you or your opponent
improving a 3️⃣ hand. In poker terminology, an “out” is any card that will improve a
player’s hand after the flop.
One common occurrence 3️⃣ is when a player holds two suited
cards and two cards of the same suit appear on the flop. The 3️⃣ player has four cards to a
flush and needs one of the remaining nine cards of that suit to complete 3️⃣ the hand. In
the case of a “four-flush”, the player has nine “outs” to make his flush.
A useful
shortcut to 3️⃣ calculating the odds of completing a hand from a number of outs is the
“rule of four and two”. The 3️⃣ player counts the number of cards that will improve his
hand, and then multiplies that number by four to calculate 3️⃣ his probability of catching
that card on either the turn or the river. If the player misses his draw on 3️⃣ the turn,
he multiplies his outs by two to find his probability of filling his hand on the
river.
In the 3️⃣ example of the four-flush, the player’s probability of filling the flush
is approximately 36% after the flop (9 outs x 3️⃣ 4) and 18% after the turn (9 outs x
2).
Pot Odds
Another important concept in calculating odds and probabilities is pot
3️⃣ odds. Pot odds are the proportion of the next bet in relation to the size of the
pot.
For instance, if 3️⃣ the pot isR$90 and the player must call aR$10 bet to continue
playing the hand, he is getting 9 to 3️⃣ 1 (90 to 10) pot odds. If he calls, the new pot is
nowR$100 and hisR$10 call makes up 10% 3️⃣ of the new pot.
Experienced players compare the
pot odds to the odds of improving their hand. If the pot odds 3️⃣ are higher than the odds
of improving the hand, the expert player will call the bet; if not, the player 3️⃣ will
fold. This calculation ties into the concept of expected value, which we will explore
in a later lesson.
Bad Beats
A 3️⃣ “bad beat” happens when a player completes a hand that
started out with a very low probability of success. Experts 3️⃣ in probability understand
the idea that, just because an event is highly unlikely, the low likelihood does not
make it 3️⃣ completely impossible.
A measure of a player’s experience and maturity is how
he handles bad beats. In fact, many experienced poker 3️⃣ players subscribe to the idea
that bad beats are the reason that many inferior players stay in the game. Bad 3️⃣ poker
players often mistake their good fortune for skill and continue to make the same
mistakes, which the more capable 3️⃣ players use against them.
Decisions, Not Results
One
of the most important reasons that novice players should understand how probability
functions at 3️⃣ the poker table is so that they can make the best decisions during a hand.
While fluctuations in probability (luck) 3️⃣ will happen from hand to hand, the best poker
players understand that skill, discipline and patience are the keys to 3️⃣ success at the
tables.
Conclusion
A strong knowledge of poker math and probabilities will help you
adjust your strategies and tactics during 3️⃣ the game, as well as giving you reasonable
expectations of potential outcomes and the emotional stability to keep playing
intelligent, 3️⃣ aggressive poker.
Remember that the foundation upon which to build an
imposing knowledge of hold’em starts and ends with the math. 3️⃣ I’ll end this lesson by
simply saying…. the math is essential.
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