First, for anyone who can answer this question, let me recommend the game Balatro. It’s a deck-building roguelite based on playing poker hands and it’s fantastic. It’s up for game of the year in a number of surveys.
For anyone not familiar with the game, you have a hand of eight cards with which to create poker hands. You also have the ability to discard and redraw up to five cards an average of three times.
If I have two pair but also have a three card proto-flush, I’m trying to decide the odds of pulling one or the other on a discard. I know that a full house is a better hand and therefore harder to find in general, but there are other factors. With a three flush, I can discard 5 cards but need to find two cards of the same suit. With two pair, I can discard 4 cards but only need one of four cards left in deck.
One last consideration is that you can pay multiple hands to reach a score. So by going for the flush draw, you’re throwing away two pair (at least let us assume that to be the case.) I’m only adding this point for those who might play balatro because it’s not an odds question.
Anyway, can anyone tell me the odds of completing a flush from three cards vs a full house from two pair, given the 8 card hands and ability to discard and redraw up to five cards? Let’s assume we only have one discard left. Thank you!
ISTM that the odds will depend on how many other players are partaking of the 44 cards in the deck that weren’t originally dealt to you. And where you are in terms of when you get to make your draw.
If you go for the flush, 10 of the 44 are of the same suit; you are drawing 5, trying to get at least 2 of them.
There are C(44,5) = 1,086,008 ways to draw 5 cards from 44. (C(44,5) refers to the number of “combinations” of drawing 5 things from a set of 44.)
There are:
C(10,5) = 252 ways of drawing 5 cards of the needed suit
C(10,4) x C(34,1) = 7140 ways of drawing 4 cards of the needed suit
C(10,3) x C(34,2) = 67,320 ways of drawing 3 cards of the needed suit
C(10,2) x C(34,3) = 269,280 ways of drawing 2 cards of the needed suit
The total number of draws that complete the flush is 343,992, so the probability = 343,992 / 1,086,008, or about 31.675%
On the other hand, if you go for the full house, you draw 4 cards, and at least one of them has to be one of the 4 cards you need
There are C(44,4) = 135,751 ways to draw 4 cards from the 44
It is probably easiest to determine the number of draws that do not make the full house; this is C(40,4), or 91,390, so that leaves 44,361 draws of 4 cards that make the full house (or better - you can make a four of a kind as well); the probability is 44,361 / 91,390, or 48.54%.
Thank you so much for doing that for me. Based on my experience (and I understand that the mind often plays tricks on itself when intuiting odds from experience), the odds of pulling a full house from two pair seems like a lot less than a coin flip. I routinely don’t pull the full house even with 3-4 discards.
There is a lot more to this as it’s not only about poker.
The idea is to beat a certain score each round to advance to the next. Each poker hand is worth so much. During play, though, you can add cards, remove cards, and change the value of the poker hands. Further, only the cards to make the hand are scored, so when you make two pair, you play four cards, not five, and replace them.
Jokers are the wild cards that change how points are scored. They might change the multiplier or point value counted. They can do that to all odd cards, all even cards, all face cards, or anything else.
There are some good play YouTube videos out there to get a sense of it.
I checked my numbers again - the probability of making the full house is actually 44,361 / 135,751, or 32.678%. That’s still better than going for the flush, but it’s not the 48% chance I claimed earlier.
Not quite true. Frequently you’ll want to add non-scoring cards to your play because they activate a power (e.g. stone cards add 50 chips) or you want to discard cards without using a discard action up.
Fair call out. I should have said that it isn’t five card poker hands. Only the cards for scoring are played but more cards might be played for their effects.