7 stud odds

[grv575]

Can someone walk through the math to calculate odds for hitting both a 3-flush on 3rd and a 3-flush on 4th asumming 8 players and no folds?

[grv575]

Actually is this the correct math (is there an easier way as well...using (x choose y) or something?):

{123} = 3-flush

{4}........{5}........{6}........{7}
hit
10/49 * [9/48 + (39/48*9/47) + (39/48*38/47*9/46)] =
...........here....or...here..........or..here
0.096247333345919311295285921954351

+

miss......hit
39/49 * 10/48 * [9/47 + 38/47*9/46]
......................here..or..here
0.057982027223470331703449187260473

+
miss......miss......hit
39/49 * 38/48 * 10/47 * 9/46
.................................here
0.026229964696331816722988918046405

= 0.1804593252657214597217240272604 * 100 ~= 18%

So it's 18% chance of hitting a 3-flush (streets designated by {street number}, here means you made the flush, hit means you hit the 4th to the suit).

[grv575]

Yeah I think it's right...if you followed all that here are simpler ones (still important ones) that should make it more clear:

3-flush on 4th
==============
10/48 * [9/47 + 38/47*9/46] = 0.072849213691026827012025901942646
hit......here or here

+

miss....hit
38/48 * 10/47 * 9/46 = 0.032955596669750231267345050878816
................here

= 0.10580481036077705827937095282146 ~= 10.6% ~= 1/10

4-flush
=======
{1234} = 4-flush

5 6 7
here or here or here
9/48 + 39/48*9/47 + 39/48*38/47*9/46 = 0.47161193339500462534690101757632 ~= 1/2
hit didn't didn't&didn't

4-flush on 5th
==============

9/47 + 38/47*9/46 = 0.3496762257169287696577243293247 ~= 1/3

Sorry for the bad formatting.

[sthief09]

3 flush on 3rd:

10 "spades" not in your hand
(10 - x) spades left in the deck, where x is the number of spades your opponents have face up
49 cards not in your hand
(49 - n) cards left in the deck, where n is the number of opponents

Total combinations that make a flush (4 more spades, 3 more spades, 2 more spades: C(10-x,4)+C(10-x,3)*C(49 - n - x,1)+C(10-x,2)*C(49 - n - x,2)
Total combinations: C(49-n,4)

For n = 7 and x = 2 (7 opponents with 2 of your suit showing), you get 21.6%

if you want to do it in Excel, make A2 = # of players, B2 = # of your suit exposed, and make this another cell:
=(COMBIN(10-B2,4)+COMBIN(10-B2,3)*COMBIN(49 - A2 - B2,1)+COMBIN(10-B2,2)*COMBIN(49 - A2 - B2,2))/COMBIN(49-A2,4)


on 4th street, there's 3 cards left
X = # of your suit exposed
n = total cards exposed

Total combinations for a flush = C(10-x,3)+C(10-x,2)*C(48 - n - x,1)
Total combinations = C(48-n,3)

let's say on 4th there's been 12 cards exposed and 4 spades... that comes to 7%

[sthief09]

you're ignoring your opponents' upcards

if they didn't have any upcards, you'd have a 28% chance on 3rd street and 13% on 4th I think.

[grv575]

Ah cool that is the way I was looking for. Looks easier than using bayesian math (conditional probabilities) which is what I attempted to do. I think my way is correct as well but it _ignores_ dead cards (and I was also using 7 other opponents). IOW it account for an average # of dead cards falling each street...so on 3rd street you have an average of 7/4 dead cards. The opponents upcards I just treated as unseen cards so it should work correctly. The combinatorics excel method doesn't handle fractions of course which is why the %s vary slightly. But if you use the figures I posted then you can just mentally adjust up or down based on a large/small # of dead cards seen per players in the hand...

Edit: Actually I don't think I did the 3-flush on 4th street properly...the % is a bit low. Have to work out the dependent odds more carefully but I like this way since it gives an average figure you can memorize and ignore other details for the moment and just adjust for live cards and live players in your head. Using your excel formula it's clear that the %'s vary drastically between 0, 1, and 2 dead cards. In fact this probably affects your implied odds so much so that it pretty much dictates if you should fold on 4-th street, having not hit the 4-th suit by how many live cards (and of course with enough players in to pay you off), or see a cheap card to try for a 4-flush draw on 5-th (only a 1/3 shot once you get that 4-flush draw).