7 stud odds

can someone help me with odds of drawing a flush.
7 card stud.
after deal, 3 spades.
no spades showing. 10 spades open.
what are the odds of making flush?

what if i have four spades and only 5 are live?

what is the formula?

thank you.

ddben

While someone holding a spade certainly changes the chance of drawing one, unless you KNOW someone has one, you still mathematically have a chance to draw all that you haven't seen.


The odds of being dealt one to a four-flush is simple:

Spades_Remaining / Cards_Remaining * Cards_Drawn


The odds of drawing two when you're holding three is trickier. If you're only drawing two more cards, the formula is:

Spades_Remaining / Cards_Remaining * ( Spades_Remaining - 1 ) / ( Cards_Remaining - 1 )

If you're drawing extra cards, then the formula is much more complex.

I *believe* if you're drawing 3 cards, the formula is:
The second formula above * 3

I *believe* if you're drawing 4 cards, the formula is:
The second formula above * 6

I could be wrong about those (It's past my bedtime). Plus, these formulas do not exclude the possibility of getting more than 5 spades.

Hmmm....

I'm already giving this a re-think.

I believe "Remaining" in all the formulas should actually be "Unseen".

I.E. You count ALL the cards, except the face up cards and your hole cards.

Actually, the math is a little more complex than that.

On the four-flush example, let's say you have a spade four-flush on fourth street. Now you have three more rounds to hit your flush. Now, IGNORING ALL OTHER UPCARDS, you have nine spades out of 48, so your probability is 18.75%. If you don't hit on fifth street, you have nine out of 47 to hit on sixth street or 19.15%. But you also have to factor in the odds of missing on fifth street. 39 out of 48 cards miss on fifth, or 81.25%. So not you have to multiply the 19.15% by 81.25%, so the actual odds of hitting specifically on sixth street are 15.56%. The math is similar on seventh street (9 of 46 to hit, given that you miss on both fifth and sixth, which is 39 of 48 and 38 of 47), giving 12.85%. So on fifth street, your probability of hitting your flush by the end is 18.75% + 15.56% + 12.85% = 47.16%. On sixth, you can get rid of the 39 of 48 on both sixth and seventh, so you get 19.15% + 15.82% = 34.87%. On seventh, you're 19.57% to hit.

Now, let's say you're up against two players. Neither player gets dealt a spade on any upcard. On fifth street, you know eight cards (your four, and each opponent's two upcards). So now, you have 9 out of 44 cards that can hit (20.45%), and 35 that will miss (39 non-flush minus the four cards your opponents have, for 79.55%). To hit on sixth, you have 9 of 41 to hit (remember, neither opponent is getting a spade) and 32 of 41 to miss (21.95% and 78.05%). On seventh, you have 9 of 38 to hit (23.68%). So now your odds to hit by the end are 20.45% + (21.95% * 79.55%) + (23.68% * 79.55% * 78.05%) = 52.61% on fifth street, 40.44% on sixth, and 23.68% on seventh.

Oh, wait, did I mention we're actually full ring, and you've actually seen the other five door cards? And that two more folded on fourth street, but you saw their fourth street cards as well? Because that changes the odds as well, significantly in your favor if none of them were spades. Now you see why nobody analyzes stud like they do Texas hold'em or draw (or if they do, I've not seen it), because it's such a goddamned pain in the ass situation specific game. Whether a player stays in or folds in draw or hold'em doesn't matter, because their cards are unknown. Since their cards are unknown, they don't affect your odds too significantly. If you're drawing one to a four-flush in draw, you know five cards, and you know nine of the 47 unknown cards are spades, so your odds are always 19.15%. Similar deal in Texas hold'em on the turn; you know five cards (your hole cards and the flop), and you're 19.15% to hit on the turn, and 34.97% to hit by the river. Stud depends on how many upcards got dealt out and how many of them were already spoken for, and this changes from hand to hand.

tl;dr:
Math is hard.
Analyzing stud sucks harder.

Oh, I suppose I should talk about the three-flush as well. The thing with hitting your flush from a three-flush is that you have to go runner-runner to do it. If you're dealt three spades as your first three cards, to get a spade on fourth and a spade on fifth means you have to get 10 of 49 AND 9 of 48, so your actual odds of doing it this way are 3.83%. Ignoring all other upcards, your odds of hitting the runner-runner flush are about 19%. Again, the other upcards can have a significant impact on your odds.

Thank you DJ.

Yes, I am counting the other upcards and know their suit, so it is not a completely unknown deck.
In your second example, if i have 3 spades to start and no other spades are showing, in an 8-way ring game, with everyone's door car showing and my two down cards, i have seen 10 cards and three of them are spades.

The real question, in 7 stud, is what are my chances of flushing on the next four draws, needing two more spades?

Thanks again!

ddben

Another Seven stud question:

once in every few hundred hands i am dealt rolled up set.
with four draws coming, what are the chances that I will hit a pair and boat up?

Thanks for your help!!

ddben

WILDQAT -

"Now you see why nobody analyzes stud like they do Texas hold'em or draw (or if they do, I've not seen it), because it's such a goddamned pain in the ass situation specific game."

this is why i play stud8. it is much more interesting and you have so much more information than you do in holdem. the odds and situations in holdem are automatic, and after a week you can make the calculations and see the situations immediately. stud8 has so many other variables to consider, especially when you throw in hi/lo 8 minimum and the chances of hitting your low to split the pot and or scoop the pot.

i feel in stud/8 that you have to play your cards much more and pay more attention to what is out on the table, while holdem (no limit) is predominantly playing the other player and also betting your whole stack on a 'race'. it seems that it is very rare that you are 'made' in a holdem game.

while we're at it, can you help me calculate the odds on:
7 stud. rolled up set. what are the odds of hitting a pair on four draws for boat? the odds of hitting the case card for quads? also, for betting purposes, do you feel it is better to just take every bet when you are rolled up, or do you ever slow play a set?

thanks.

ddben