In the previous article on working out preflop hand probability, we worked out the likelihood of being dealt different combinations of starting hands before the flop.
- Poker Odds Chart
- Poker Preflop Equity Chart
- How To Calculate Poker Equity Preflop
- Texas Holdem Odds Chart Preflop
If the how to calculate poker odds preflop dealer is showing a 4, 5, or 6 card, they have more of a chance of going over the 21 limit. Poker Odds Tell You the Probability of Winning Any Given Hand. Before we can get into a discussion of poker odds while playing poker online, you need to know how to calculate your 'outs.' PokerListings.com’s Poker Odds Calculator is the fastest, most accurate and easy-to-use poker odds calculator online. It’s just like what you see when you watch poker on TV. Use it in real-time to know exactly what your chances of winning and losing are at any point in a poker hand – be it on online poker sites or playing live poker.
In this article, I will cover the basics of working out the probabilities behind various possible flops. I'll go ahead and cover the probability basics first in case you missed it in the preflop probability article.
- Probability calculations quick links.
A few probability basics.
When working out flop probabilities, the main probabilities we will work with are the number of cards left in the deck and the number of cards we want to be dealt on the flop. So for example, if we were going to deal out 1 card:
- The probability of dealing a 7 would be 1/52 - There is one 7 in a deck of 52 cards.
- The probability of dealing any Ace would be 4/52 - There four Aces in a deck of 52 cards.
- The probability of dealing any would be 13/52 - There are 13 s in a deck of 52 cards.
In fact, the probability of being dealt any random card (not just the 7) would be 1/52. This also applies to the probability being dealt any random value of card like Kings, tens, fours, whatever (4/52) and the probability of being dealt any random suit (13/52).
Each card is just as likely to be dealt as any other - no special priorities in this game!
The numbers change for future cards.
A quick example... let's say we want to work out the probability of being dealt a pair of sevens.
- The probability of being dealt a 7 for the first card will be 4/52.
- The probability of being dealt a 7 for the second card will be 3/51.
Notice how the probability changes for the second card? After we have been dealt the first card, there is now 1 less card in the deck making it 51 cards in total. Also, after already being dealt a 7, there are now only three 7s left in the deck.
Always try and take care with the numbers for future cards. The numbers will change slightly as you go along.
Working out probabilities.
- Whenever the word 'and' is used, it will usually mean multiply.
- Whenever the word 'or' is used, it will usually mean add.
This won't make much sense for now, but it will make a lot of sense a little further on in the article. Trust me.
Total number of flop combinations.
First of all, lets work out the total number of possible flop combinations. In other words, we will just be working out the probability of 'any random flop'.
To work out this probability we simply multiply the probability of 3 individual cards being dealt.
- (random card) * (random card) * (random card)
- (1/52) * (1/51) * (1/50) = 132,600 possible flops.
Pretty big combination of cards huh? However, we've omitted the fact that we know our 2 holecards, so there will be two less known cards in the deck when we are dealing the flop. So if we amend this calculation by starting at 50 instead of 52:
P = (1/50) * (1/49) * (1/48)
P = 1/117,600.
Better, but this 1/117,600 probability is with exact cards in order. In Texas Hold'em it does not make a difference whether the flop comes A K Q or A Q K. So to account for this we multiply this fraction by 6 (1*2*3 = 6).
P = 1/117,600 * 6
P = 1/19,600.
The order of cards on the flop makes no difference, so multiply the probability by 6 to account for this (1 * 2 * 3 = 6 - this is math probability stuff). Don't worry if you don't know why we do this, just take it as it is.
This means that the probability of the flop being A K Q in any order is 1/19,600 - which is exactly the same probability as the flop coming something like 2 5 9 in any order.
So in total there are 19,600 different possible flops in Texas Hold'em.
Probability of specific flops.
To work out the probability of specific flops with the cards in any order we simply multiply the probabilities of each of those cards being dealt.
Multiply the 3 probabilities together.
So let's say we want to find the probability of flopping a heart flush.
- There are 2 hearts in our hand.
- There are 11 hearts left in a deck of 50.
P = (11/50) * (10/49) * (9/48)
P = 990/117600 = 1/119
In this example we do not need to multiply the final probability by 6. This is because the order of the cards and their probabilities are important, as the overall probabilities decrease as each heart is dealt.
Poker Odds Chart
Overview of working out flop probabilities.
This is a really basic article for working out flop probabilities in Texas Hold'em. Think of it as more of a taster for working out probabilities on the flop to help you get your feet wet.
If I were to continue with probabilities, I would be delving deeper in to mathematics and further away from poker. That wouldn't necessarily be a bad thing, but my maths is a bit rusty and it's not going to directly improve your game, so I'll stop for now. Maybe I'll address it in a future article, but for now this is as far as I'm going to go.
If you're really interested in the mathematics of the game, try the book: The Mathematics of Poker by Bill Chen. It's definitely not a light read, but it's very interesting if you enjoy the maths side of the game. For other books, try the poker books section.
Other useful articles.
- Poker mathematics.
- Pot odds.
- Equity in poker.
Go back to the poker odds charts.
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Ever wondered where some of those odds in the odds charts came from? In this article, I will teach you how to work out the probability of being dealt different types of preflop hands in Texas Holdem.
It's all pretty simple and you don't need to be a mathematician to work out the probabilities. I'll keep the math part as straightforward as I can to help keep this an easy-going article for the both of us.
- Probability calculations quick links.
A few probability basics.
When working out hand probabilities, the main probabilities we will work with are the number of cards in the deck and the number of cards we want to be dealt. So for example, if we were going to deal out 1 card:
- The probability of dealing a 7 would be 1/52 - There is one 7 in a deck of 52 cards.
- The probability of dealing any Ace would be 4/52 - There four Aces in a deck of 52 cards.
- The probability of dealing any would be 13/52 - There are 13 s in a deck of 52 cards.
In fact, the probability of being dealt any random card (not just the 7) would be 1/52. This also applies to the probability being dealt any random value of card like Kings, tens, fours, whatever (4/52) and the probability of being dealt any random suit (13/52).
Each card is just as likely to be dealt as any other - no special priorities in this game!
The numbers change for future cards.
A quick example... let's say we want to work out the probability of being dealt a pair of sevens.
- The probability of being dealt a 7 for the first card will be 4/52.
- The probability of being dealt a 7 for the second card will be 3/51.
Notice how the probability changes for the second card? After we have been dealt the first card, there is now 1 less card in the deck making it 51 cards in total. Also, after already being dealt a 7, there are now only three 7s left in the deck.
Always try and take care with the numbers for future cards. The numbers will change slightly as you go along.
Working out probabilities.
- Whenever the word 'and' is used, it will usually mean multiply.
- Whenever the word 'or' is used, it will usually mean add.
This won't make much sense for now, but it will make a lot of sense a little further on in the article. Trust me.
Probability of being dealt two exact cards.
Multiply the two probabilities together.
So, we want to find the probability of being dealt the A and K. (See the 'and' there?)
- Probability of being dealt A - 1/52.
- Probability of being dealt K - 1/51.
Now let's just multiply these bad boys together.
P = (1/52) * (1/51)
P = 1/2652
So the probability of being dealt the A and then K is 1/2652. As you might be able to work out, this is the same probability for any two exact cards, as the likelihood of being dealt A K is the same as being dealt a hand like 7 3 in that order.
But wait, we do not care about the order of the cards we are dealt!
When we are dealt a hand in Texas Hold'em, we don't care whether we get the A first or the K first (which is what we just worked out), just as long as we get them in our hand it's all the same. There are two possible combinations of being dealt this hand (A K and K A), so we simply multiply the probability by 2 to get a more useful probability.
P = 1/2652 * 2
P = 1/1326
You might notice that because of this, we have also worked out that there are 1,326 possible combinations of starting hands in Texas Holdem. Cool huh?
Probability of being dealt a certain hand.
Two exact cards is all well and good, but what if we want to work out the chances of being dealt AK, regardless of specific suits and whatnot? Well, we just do the same again...
Multiply the two probabilities together.
So, we want to find the probability of being dealt any Ace andany King.
- Probability of being dealt any Ace - 4/52.
- Probability of being dealt any King - 4/51 (after we've been dealt our Ace, there are now 51 cards left).
P = (4/52) * (4/51)
P = 16/2652 = 1/166
However, again with the 2652 number we are working out the probability of being deal an Ace and then a King. If we want the probability of being dealt either in any order, there are two possible ways to make this AK combination so we multiply the probability by 2.
P = 16/2652 * 2
P = 32/2652
P = 1/83
The probability of being dealt any AK as opposed to an AK with exact suits is more probable as we would expect. A lot more probable in fact. Also, as you might guess, this probability of 1/83 will be the same for any two value of cards like; AQ, JT, 34, J2 and so on regardless of whether they are suited or not.
Probability of being dealt a range of hands.
Work out each individual hand probability and add them together.
Poker Preflop Equity Chart
What's the probability of being dealt AA or KK? (Spot the 'or' there? - Time to add.)
How To Calculate Poker Equity Preflop
- Probability of being dealt AA - 1/221 (4/52 * 3/51 = 1/221).
- Probability of being dealt KK - 1/221 (4/52 * 3/51 = 1/221).
P = (1/221) + (1/221)
P = 2/221 = 1/110
Easy enough. If you want to add more possible hands in to the range, just work out their individual probability and add them in. So if we wanted to work out the odds of being dealt AA, KK or 7 3...
- Probability of being dealt AA - 1/221 (4/52 * 3/51 = 1/221).
- Probability of being dealt KK - 1/221 (4/52 * 3/51 = 1/221).
- Probability of being dealt 7 3 - 1/1326 ([1/52 * 1/51] * 2 = 1/1326).
P = (1/221) + (1/221) + (1/1326)
P = 359/36465 = 1/102
Texas Holdem Odds Chart Preflop
This one definitely takes more skill with adding fractions because of the different denominators, but you get the idea. I'm just teaching hand probabilities here, so I'm not going to go in to adding fractions in this article for now! This fractions calculator is really handy for adding those trickier probabilities quickly though.
Overview of working out hand probabilities.
Hopefully that's enough information and examples to allow you to go off and work out the probabilities of being dealt various hands and ranges of hands before the flop in Texas Holdem. The best way to learn how to work out probabilities is to actually try and work it out for yourself, otherwise the maths part will just go in one ear and out the other.
I guess this article isn't really going to do much for improving your game, but it's still pretty interesting to know the odds of being dealt different types of hands.
I'm sure that some of you reading this article were not aware that the probability of being dealt AA were exactly the same as the probability of being dealt 22! Well, now you know - it's 1/221.
Other useful articles.
- Poker mathematics.
- Pot odds.
- Equity in poker.
Go back to the poker odds charts.
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