Texas Hold’em Poker Hand Odds - A Primer
Understanding hand odds should be a crucial part of your strategy for playing Texas Hold’em, or any of its variations. It’s not exactly easy to think on the fly, especially when it comes to dividing strange fractions in your head during a tournament. So you should get a handle on how to calculate when you’re not playing.
Called me biased - I’m a math geek - but it’s not too difficult to calculate approximate odds and other important figures if you have a basic understanding of how to do it. Some of this stuff will be obvious, but for the sake of argument, I’ll assume you don’t know how to calculate any card odds. I’ve talked about odds before, but I wanted to try to bring some basics together in a single post. (Warning: this is a long post.)
Firstly, in a regular deck of cards, there are 4 suits and 13 cards in each suit. Each suit has 3 face cards (royalty), but there are 5 cards which have a value of 10 in poker: A through 10. Keep in mind that this is different from Blackjack, where A is either 1 or 11.
Holding any two hole cards (hidden from other players) that are both worth 10 points each is called holding 20 points. Any time you are holding 20 in the hole in Texas Hold’em, you are holding a relatively strong hand that you should consider playing. Personally, I play tight and conservative, so I won’t play just any 20 hand.
Since each player in a game is dealt two cards at a time, you can calculate the approximate odds of receiving a particular duo of cards fairly easily. For example, say that you are the first player to be dealt your two cards and you want to determine the odds that you will be dealt a 20 hand. How do we figure that out?
Simple. First, how many 10-pt cards are there in a deck? There are 5 in each suit, and 4 suits, for a total of 20 in a deck of 52. So your odds are 20/52 =~ 0.3846 = 38.46 %. So better than 1/3 odds of being dealt a strong hand.
What about the second player to be dealt? What are their odds? Well, here’s where it gets a little confusing. It depends on who is calculating. If you are the first player, or the commentator in a live tourney, you know the first 2 cards dealt. The second player does not, and has to calculate their odds the same way as the first player did.
The odds from the omnipotent perspective (i.e., the real one) would be as follows. Say two 10 pt cards have been dealt to the first player. The second player then has (20-2)/ (52-2) =~ 0.36 = 36.00% odds of also getting a 20-hand, in reality. But they don’t know that. They either have to assume 38.46%, as with the first player, or better. For example, if the first player did not get a 20, then the second player has 20/(52-2) = 40.00% odds of getting a 20. So the easiest, as a player, is to assume the worst case scenario, 36%, and then try to guess what the other player holds.
Now that you know the difference between real odds and perspective odds, you can figure out what the odds are for each successive player in a match to get a 20 hand. Obviously, since there 20 10-pt cards in a deck and no more than 10 players in a game of Texas Hold’em, they could conceivably all have 20-hands.
What are the odds of that happening, though? I leave that as an exercise for you, but it’s very small. (Hint: multiply the real odds for each player.)
What about the player odds you see on TV, during a Texas Hold’em tourney? Those are “real” odds, calculated only because the commentators (and audience) know exactly what every player is holding. We have an advantage that obviously the players do not.
So how do they calculate the real odds of a player winning a hand? Let’s talk post-flop, as it’s way too complicated to talk pre-flop just yet. Recall the rules of Texas Hold’em. Post-flop occurs after “burn one, turn three.” That is, after the players have been dealt their two cards apiece, and the first betting round ends, the dealer “burns” the top card. That is, puts it aside, face down. Then three community cards are revealed.
At this point, it’s likely every player’s hand odds have changed, both from their perspective and from the omniscient perspective. There is no way you can generalize a formula at this point. You have to calculate your own odds or the omniscient odds on a case by case basis.
Let’s say that you have Ad-10c, and the flop is 3s-2d-6h. What are your odds of winning? Well, in this case, pretty low. But let’s be specific. Regardless of what other players have, what cards on 4th and 5th street would help you? Any 4 AND any 5 cards would give you a straight, but every one in the game would have that as well. If no one gets anything out of 4th and 5th street, you do have Ace-high, and you can hope that no one else has an Ace.
With this example, you cannot get a straight that would help just you. So there are only a few ways that you can get a strong hand: have one or more Aces or tens show on 4th and 5th streets. What are the chances of that happening? Well, from your perspective, you’ll have to determine best and worst-case scenarios.
Best-case: No one else has an A or 10 in their hand, and it isn’t the one burned card. How many players are there? Say 5. Each one has 2 cards. So the deck after the flop has 52 - 1 - (5×2) = 41 cards. Don’t forget that before 4th street is dealt, one more card is burnt first. So assuming you are the only person with A or 10, the chances of flopping either are as follows:
A: There are 3 Aces left, so 3/(41-1) =~ 0.0750 = 7.50%
10: There are 3 Tens left, so 7.50%
Chances flopping either on 4th Street: (3+3)/(41-1) =~ 0.1500 = 15.00%. And that’s the best case scenario.
Worst-case: All the Aces and Tens are either held by other players or one is the burned card. So your odds are 0/40 = 0%. But what are the odds of that happening? Well, since you have an A-10, there are only 6 left of either, after you’ve been dealt your hand. If you were the first person to be dealt a hand, then the second person could have either one A or 10, or two A, two 10, or A-10. You have to calculate the odds on each of these three possibilties and add them up. Not worth it, is it? Just go with 0%, in this example.
Real-odds: Your real odds of flopping A or 10 on 4th Street are somewhere between best- and worst-case scenario: 0-15%. The real odds can only be calculated by someone who knows what the other players hold, but from a player perspective, this is enough to make a bet or pass, depending on what you think the other players have.
Those are the basics, using A-10 in the first player’s hand as an example. There are more complicated odds that can be calculated, especially omniscient odds. But from each player’s perspective only their best- and worst-case scenarios matter. You can estimate your odds ahead of time of also getting a favorable card in 5th street, then decide now whether you want to continue betting, or folding.
At some later date, I may attempt to explain how to calculate odds pre-flop.
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