While it's tempting to think that poker is all about quads against straight flushes for millions of dollars, most of the time you're going to have a weak holding such as a pair or just a high card. But just how common are the least valuable poker hands?
First, we need to understand some basic poker maths.
Poker Math Primer
In order to see just how common the worst poker hands really are, we need to understand how we work out the probability of making hands in poker (i.e., the odds).
The way you work out the odds will change depending on the variant of poker you play as the number of cards you have will change from one to the next. The most basic game in terms of math is 5 card draw so we'll start with that.
5 Card Draw
5 card draw is the most basic as you are given the absolute minimum cards to make a poker hand (a poker hand being made up of 5 cards), whereas games like Omaha, Holdem, Stud, etc. give you more than five and you pick your five best to make your hand.
To work out the odds of each hand type we first need to work out the total number of card combinations.
The total number of cards that we can choose from to start with is 52 so the probability of any card being our first card is 1 out of 52, for the second card there are 51 cards left so the probability of a card being our second card is 1 out of 51, from there it's 1 out of 50, 1 out of 49, and 1 out of 48 for our third, fourth, and fifth cards.
To get the total number of combinations we multiply these numbers together:
52*51*50*49*48 = 311,875,200
That's a lot of combinations! But it's a little misleading. As the order of the cards doesn't matter we need to remove the duplicate combinations and to do so we divide our original number by 120:
311,875,200 / 120 = 2,598,960 - The total hand combinations for 5 card draw
For Texas Hold'em, we have 7 cards to choose from rather than 5: the two hole cards and the five community cards. This means that the first equation becomes a lot bigger:
52*51*50*49*48*47*46 = 674,274,182,400
But similarly to the 5 card draw, we need to take into account duplicate combinations. The number of which is also significantly greater so the number we need to divide our original number by is 5,040:
674,274,182,400 / 5040 = 133,784,560 - The total number of 7 card combinations
5 Lowest Poker Hands According to Math
According to occurrence probabilities, these are the lowest-ranked poker hands you can have.
High card is the lowest-ranked poker hand there is, it's any hand that doesn't have a pair or better. The odds of being left with just a high card are substantially different in 5 card draw compared to Texas Hold'em. Having the extra two cards in Hold'em makes it a lot harder for there to not be a pair out there.
The mathematical proof for the probability for no pair in 5 card draw is as follows:
P(high - card) =1,317,888 −10,200 − 5,108 − 36 − 42,598,960 = 50.11%
Around half the time you will be left without a pair! Now let's compare that to the probability of no pair in Texas Hold'em:
P(high - card) = 1499 * (47 - 756 - 4 - 84)
This is equal to 23,294,460 hand combos or around 17.4% of total hands - a big drop off compared to 5 card draw!
One pair can be a hand that you're happy to get a lot of money in with - especially if it's a pair of aces! However, it's still the second-worst hand ranking there is and is one of the most common hands you'll have. The probabilities of these hands occurring in 5 card draw and Hold'em are very similar, let's look at 5 card draw first:
P(1 pair) = 1,098,240 / 2,598,960 = 42.25%
This is equal to around 42.25% of the time, more common than people might think. What's even more surprising is how often it occurs in Texas Hold'em:
P(1 pair) = [(13/6) - 71] 6 * 6 * 990
Which is equal to 43.8% of the time! For anyone who has played Texas Hold'em, that seems like a lot, but it's worth remembering that this includes pairs on the board, not just pairs that we make using our hand.
This is where we start to see a big difference between 5 card draw and Texas Hold'em. Given that we only get five cards and making two-pair involves four cards, it's a very hard hand to make. Compare this to Texas Hold'em, where we now have seven cards to use instead of five, we're much more likely to make hands that involve more cards. Let's see how this plays out in the math:
P(2 pair) = 123,552 / 2,598,960 = 4.75%
This is a big drop-off compared to the one pair hands! But it makes sense when we remember that it takes 80% of the available cards to make this hand. The more cards we have to choose from, the easier it is to make these hands as we'll see with Texas Hold'em:
P(2 pair) = [1277 * 10 * [6 * 62 + 24 * 63 + 6 * 64]] + [(13/3) (4/2)^3 (40/1)]
Which is equal to 23.5%! Only about half as likely as one pair! Again to long-time Texas Hold'em players, this will seem incorrect but it includes pairs on the board. If we limited it to making two-pair using the cards from your hand that percentage would be a lot lower.
Three of a Kind
We're moving up in the hand rankings now as three of a kind is generally considered a strong hand in both games. Therefore we're going to see hands from now on become harder and harder to make compared to the relative ease of making one pair:
P(3 of a kind) = 54,912 / 2,598,960 = 2.11%
Again, when playing 5 card draw any hand that requires more than half the available cards is going to be hard to make and that's pattern repeats itself with the likelihood of making three of a kind. Given the drop off between making one pair and two pair, this doesn't seem as much of a difference but it's still a tough hand to make.
P(3 of a kind) = [(13/5) - 10] * (5/1) * (4/1) * [(4/1)^4 - 3]
In Texas Hold'em the probability isn't quite as low as 5 card draw at 4.83%. Having those extra cards makes a lot of difference when it comes to these higher ranked hands and it's a pattern you'll see continued if you look further up the hand ranking charts.
Even though a straight isn't considered to be a weak poker hand, it's still the 5th lowest hand ranking there is. Given that you need to use five cards to make a straight we can assume that these hands are going to have a very low probability for 5 card draw:
P(straight not flush) = (10,240 − 36 − 4) / 2,598,960 = 0.76%
And we have been proven right! You're going to make a straight less than 1% of the time playing 5 card draw which isn't very often at all. If it's this low for making a straight, just think how hard it is to make hands like flushes and full houses.
P(straight not flush) = [217 * [47 - 756 - 4 - 84]] + [71 * 36 * 990] + [10 * 5 * 4 * [256 - 3] + 10 * (5/2) * 2268]
Blimey, that's a long equation! In case you can't figure this one out, it's equal to around 4.62% of the time, very similar to the probability of making three of a kind. Having those extra cards is so important - if you think these hands are common in Texas Hold'em, imagine how common they are in a game like 6-card PLO!
Now that we know just how likely you are to have these weak hands when you're playing, it's important that you study how best to play them as more often than not these are the hands you'll have to play!
This article was published on April 4, 2021, and last updated on April 4, 2021.