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The main underpinning of poker is math – it is essential. For every decision you make,
while factors such as 💵 psychology have a part to play, math is the key element.
In this
lesson we’re going to give an overview of 💵 probability and how it relates to poker. This
will include the probability of being dealt certain hands and how often 💵 they’re likely
to win. We’ll also cover how to calculating your odds and outs, in addition to
introducing you to 💵 the concept of pot odds. And finally we’ll take a look at how an
understanding of the math will help 💵 you to remain emotional stable at the poker table
and why you should focus on decisions, not results.
What is Probability?
Probability 💵 is
the branch of mathematics that deals with the likelihood that one outcome or another
will occur. For instance, a 💵 coin flip has two possible outcomes: heads or tails. The
probability that a flipped coin will land heads is 50% 💵 (one outcome out of the two);
the same goes for tails.
Probability and Cards
When dealing with a deck of cards the
💵 number of possible outcomes is clearly much greater than the coin example. Each poker
deck has fifty-two cards, each designated 💵 by one of four suits (clubs, diamonds, hearts
and spades) and one of thirteen ranks (the numbers two through ten, 💵 Jack, Queen, King,
and Ace). Therefore, the odds of getting any Ace as your first card are 1 in 13 💵 (7.7%),
while the odds of getting any spade as your first card are 1 in 4 (25%).
Unlike coins,
cards are 💵 said to have “memory”: every card dealt changes the makeup of the deck. For
example, if you receive an Ace 💵 as your first card, only three other Aces are left among
the remaining fifty-one cards. Therefore, the odds of receiving 💵 another Ace are 3 in 51
(5.9%), much less than the odds were before you received the first Ace.
Pre-flop
Probabilities: 💵 Pocket Pairs
In order to find the odds of getting dealt a pair of Aces,
we multiply the probabilities of receiving 💵 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 💵 local
casino and are dealt 30 hands per hour, you can expect to receive pocket Aces an
average of once 💵 every 7.5 hours.
The odds of receiving any of the thirteen possible
pocket pairs (twos up to Aces) is:
(13/221) = (1/17) 💵 ≈ 5.9%.
In contrast, you can
expect to receive any pocket pair once every 35 minutes on average.
Pre-Flop
Probabilities: Hand vs. 💵 Hand
Players don’t play poker in a vacuum; each player’s hand
must measure up against his opponent’s, especially if a player 💵 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 💵 the chances of certain events
occurring when playing certain starting hands. The following table lists some
interesting and valuable hold’em 💵 math:
Many beginners to poker overvalue certain
starting hands, such as suited cards. As you can see, suited cards don’t make 💵 flushes
very often. Likewise, pairs only make a set on the flop 12% of the time, which is why
small 💵 pairs are not always profitable.
PDF Chart
We have created a poker math and
probability PDF chart (link opens in a new 💵 window) which lists a variety of
probabilities and odds for many of the common events in Texas hold ’em. This 💵 chart
includes the two tables above in addition to various starting hand probabilities and
common pre-flop match-ups. You’ll need to 💵 have Adobe Acrobat installed to be able to
view the chart, but this is freely installed on most computers by 💵 default. We recommend
you print the chart and use it as a source of reference.
Odds and Outs
If you do see 💵 a
flop, you will also need to know what the odds are of either you or your opponent
improving a 💵 hand. In poker terminology, an “out” is any card that will improve a
player’s hand after the flop.
One common occurrence 💵 is when a player holds two suited
cards and two cards of the same suit appear on the flop. The 💵 player has four cards to a
flush and needs one of the remaining nine cards of that suit to complete 💵 the hand. In
the case of a “four-flush”, the player has nine “outs” to make his flush.
A useful
shortcut to 💵 calculating the odds of completing a hand from a number of outs is the
“rule of four and two”. The 💵 player counts the number of cards that will improve his
hand, and then multiplies that number by four to calculate 💵 his probability of catching
that card on either the turn or the river. If the player misses his draw on 💵 the turn,
he multiplies his outs by two to find his probability of filling his hand on the
river.
In the 💵 example of the four-flush, the player’s probability of filling the flush
is approximately 36% after the flop (9 outs x 💵 4) and 18% after the turn (9 outs x
2).
Pot Odds
Another important concept in calculating odds and probabilities is pot
💵 odds. Pot odds are the proportion of the next bet in relation to the size of the
pot.
For instance, if 💵 the pot isR$90 and the player must call aR$10 bet to continue
playing the hand, he is getting 9 to 💵 1 (90 to 10) pot odds. If he calls, the new pot is
nowR$100 and hisR$10 call makes up 10% 💵 of the new pot.
Experienced players compare the
pot odds to the odds of improving their hand. If the pot odds 💵 are higher than the odds
of improving the hand, the expert player will call the bet; if not, the player 💵 will
fold. This calculation ties into the concept of expected value, which we will explore
in a later lesson.
Bad Beats
A 💵 “bad beat” happens when a player completes a hand that
started out with a very low probability of success. Experts 💵 in probability understand
the idea that, just because an event is highly unlikely, the low likelihood does not
make it 💵 completely impossible.
A measure of a player’s experience and maturity is how
he handles bad beats. In fact, many experienced poker 💵 players subscribe to the idea
that bad beats are the reason that many inferior players stay in the game. Bad 💵 poker
players often mistake their good fortune for skill and continue to make the same
mistakes, which the more capable 💵 players use against them.
Decisions, Not Results
One
of the most important reasons that novice players should understand how probability
functions at 💵 the poker table is so that they can make the best decisions during a hand.
While fluctuations in probability (luck) 💵 will happen from hand to hand, the best poker
players understand that skill, discipline and patience are the keys to 💵 success at the
tables.
Conclusion
A strong knowledge of poker math and probabilities will help you
adjust your strategies and tactics during 💵 the game, as well as giving you reasonable
expectations of potential outcomes and the emotional stability to keep playing
intelligent, 💵 aggressive poker.
Remember that the foundation upon which to build an
imposing knowledge of hold’em starts and ends with the math. 💵 I’ll end this lesson by
simply saying…. the math is essential.
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Zucman disse que valorizar a riqueza dos bilionários seria relativamente simples porque grande parte dela foi 🍎 realizada na forma de ações. Seu relatório diz poder ser aplicado com sucesso mesmo se todos os países não o 🍎 adotassem, fortalecendo impostos atuais sobre saída (leves novibet ios pessoas ricas levando seu dinheiro para uma jurisdição sem participação) e implementam 🍎 "coletora do imposto da última instância".
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A proposta será discutida na reunião do G20 de ministros das Finanças no Rio.
O diretor executivo interino da 🍎 Organização Internacional de Energia e Desenvolvimento Econômico (Oxfam), Amitabh Behar, disse: "Esta é uma proposta sensata que está no interesse 🍎 econômico estratégico do governo. Todos os países G20 devem apoiar o esforço brasileiro para garantir a primeira transação global com 🍎 impostos sobre super-ricos."
Phil White, membro da Patriotic Millionaires UK disse: “O relatório de Zucman mostra que uma abordagem coordenada e 🍎 internacional para taxar os super-ricos é a maneira sensata E justa Para lidar com as concentrações extremas. riqueza ”.
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Estender 🍎 o plano para aqueles com riqueza superior a USR$ 100 milhões ( 79m) levantaria um adicional de R$100bn-140 bilhões por 🍎 ano, disse Zucman.
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