Pai Gow Probability question

According to Wizard's Fortune Pai Gow page, there are 6,172,088 combinations out of 154,143,080 that make a Flush, for odds of 4.004%.

How do I calculate how many of those combinations contain either an Ace or a Joker or both? Or simply put, what are the odds of having an Ace-High Flush in Pai Gow?

To be clear, I don't want to count Straight Flushes and Royal Flushes in this total.

Thanks!

Quote: Deucekies

According to Wizard's Fortune Pai Gow page, there are 6,172,088 combinations out of 154,143,080 that make a Flush, for odds of 4.004%.

How do I calculate how many of those combinations contain either an Ace or a Joker or both? Or simply put, what are the odds of having an Ace-High Flush in Pai Gow?

To be clear, I don't want to count Straight Flushes and Royal Flushes in this total.

Thanks!
link to original post

I have spreadsheets I could interrogate to get an answer, but I think I can calculate this using combination math. The trick is to do the calculations assuming there are only 12 ranks (which corresponds to flushes without an Ace or Joker).

Straight flushes with 12 ranks
A= 7 card SF = 7*4
B= 6 card SF = 7*4*39
C= 5 card SF = 8*4*combin(39,2)

Flushes with 12 ranks
D= 7 card Flush = combin(12,7)*4
E= 6 card flush = combin(12,6) *combin(36,1)
F=5 card flush = combin (12,5)* combin (36,2)

So, flush combinations without an Ace or Joker = G = D+E+F-A-B-C

(I did this quickly and then had to leave for dinner, so it may have msitakes. Review by others would be appreciated.)

Quote: gordonm888

Quote: Deucekies

According to Wizard's Fortune Pai Gow page, there are 6,172,088 combinations out of 154,143,080 that make a Flush, for odds of 4.004%.

How do I calculate how many of those combinations contain either an Ace or a Joker or both? Or simply put, what are the odds of having an Ace-High Flush in Pai Gow?

To be clear, I don't want to count Straight Flushes and Royal Flushes in this total.

Thanks!
link to original post

I have spreadsheets I could interrogate to get an answer, but I think I can calculate this using combination math. The trick is to do the calculations assuming there are only 12 ranks (which corresponds to flushes without an Ace or Joker).

Straight flushes with 12 ranks
A= 7 card SF = 7*4
B= 6 card SF = 7*4*39
C= 5 card SF = 8*4*combin(39,2)

Flushes with 12 ranks
D= 7 card Flush = combin(12,7)*4
E= 6 card flush = combin(12,6) *combin(36,1)
F=5 card flush = combin (12,5)* combin (36,2)

So, flush combinations without an Ace or Joker = G = D+E+F-A-B-C

(I did this quickly and then had to leave for dinner, so it may have msitakes. Review by others would be appreciated.)
link to original post

Did you remember that a flush including a joker may be a straight flush that shouldn’t count?

Like 4,5,7,8 Joker?

Quote: SOOPOO

Quote: gordonm888

Quote: Deucekies

According to Wizard's Fortune Pai Gow page, there are 6,172,088 combinations out of 154,143,080 that make a Flush, for odds of 4.004%.

How do I calculate how many of those combinations contain either an Ace or a Joker or both? Or simply put, what are the odds of having an Ace-High Flush in Pai Gow?

To be clear, I don't want to count Straight Flushes and Royal Flushes in this total.

Thanks!
link to original post

I have spreadsheets I could interrogate to get an answer, but I think I can calculate this using combination math. The trick is to do the calculations assuming there are only 12 ranks (which corresponds to flushes without an Ace or Joker).

Straight flushes with 12 ranks
A= 7 card SF = 7*4
B= 6 card SF = 7*4*39
C= 5 card SF = 8*4*combin(39,2)

Flushes with 12 ranks
D= 7 card Flush = combin(12,7)*4
E= 6 card flush = combin(12,6) *combin(36,1)
F=5 card flush = combin (12,5)* combin (36,2)

So, flush combinations without an Ace or Joker = G = D+E+F-A-B-C

(I did this quickly and then had to leave for dinner, so it may have msitakes. Review by others would be appreciated.)
link to original post

Did you remember that a flush including a joker may be a straight flush that shouldn’t count?

Like 4,5,7,8 Joker?
link to original post

In a twelve rank (2-K) analysis, none of the SFs will have been made with a joker, as will none of the ordinary flushes. And, when you take the total combinations of 12-rank flushes and subtract them from the 154,143,080 combinations for flushes that Wizard reports, then the remainder must be flushes with an Ace or a flush acting as an Ace. So. I think I'm okay. Your point is a good one though -if I were trying to calculate something equivalent for the straight flushes the issue you point out would be a major complication.

If I can throw in here, a player recently hit a Pai Gow progressive for a little south of $1.4 mil. The hand was a 7 card straight flush. (ie: straight flush in both hands.)

What were the odds against that?

Quote: AZDuffman

If I can throw in here, a player recently hit a Pai Gow progressive for a little south of $1.4 mil. The hand was a 7 card straight flush. (ie: straight flush in both hands.)

What were the odds against that?
link to original post

Take out your ‘i.e.’ it has to be a 7 card straight flush. Not a 5 card straight flush and a 2 card straight flush. 2,3,5,6,7,8,9 all hearts is NOT a winner.

Edit. I believe ams hit it once for low 6 figures?

Quote: SOOPOO

Quote: AZDuffman

If I can throw in here, a player recently hit a Pai Gow progressive for a little south of $1.4 mil. The hand was a 7 card straight flush. (ie: straight flush in both hands.)

What were the odds against that?
link to original post

Take out your ‘i.e.’ it has to be a 7 card straight flush. Not a 5 card straight flush and a 2 card straight flush. 2,3,5,6,7,8,9 all hearts is NOT a winner.

Edit. I believe ams hit it once for low 6 figures?
link to original post

OK, then, sorry if I got that wrong. This his was for 1.36 or so mil and was over a year and a half since last hit. Wondering how many hands.

I know the last part can be weird. In Texas Hold em I once was involved in two royals in a matter of a few weeks. One I got one I dealt. Second one ruined much drama for the night but that is another story, I will put it in Nathan's Corner if anyone cares to hear it.

Quote: gordonm888

Quote: SOOPOO

Quote: gordonm888

Quote: Deucekies

According to Wizard's Fortune Pai Gow page, there are 6,172,088 combinations out of 154,143,080 that make a Flush, for odds of 4.004%.

How do I calculate how many of those combinations contain either an Ace or a Joker or both? Or simply put, what are the odds of having an Ace-High Flush in Pai Gow?

To be clear, I don't want to count Straight Flushes and Royal Flushes in this total.

Thanks!
link to original post

I have spreadsheets I could interrogate to get an answer, but I think I can calculate this using combination math. The trick is to do the calculations assuming there are only 12 ranks (which corresponds to flushes without an Ace or Joker).

Straight flushes with 12 ranks
A= 7 card SF = 7*4
B= 6 card SF = 7*4*39
C= 5 card SF = 8*4*combin(39,2)

Flushes with 12 ranks
D= 7 card Flush = combin(12,7)*4
E= 6 card flush = combin(12,6) *combin(36,1)
F=5 card flush = combin (12,5)* combin (36,2)

So, flush combinations without an Ace or Joker = G = D+E+F-A-B-C

(I did this quickly and then had to leave for dinner, so it may have msitakes. Review by others would be appreciated.)
link to original post

Did you remember that a flush including a joker may be a straight flush that shouldn’t count?

Like 4,5,7,8 Joker?
link to original post

In a twelve rank (2-K) analysis, none of the SFs will have been made with a joker, as will none of the ordinary flushes. And, when you take the total combinations of 12-rank flushes and subtract them from the 154,143,080 combinations for flushes that Wizard reports, then the remainder must be flushes with an Ace or a flush acting as an Ace. So. I think I'm okay. Your point is a good one though -if I were trying to calculate something equivalent for the straight flushes the issue you point out would be a major complication.
link to original post

I literally woke up at 4:30 a.m. last night with the realization that I was wrong in the above comment (as someone has pointed out, I think) and that Soopoo was right. When subtracting out the 12-rank flush combinations from the total combinations for all flushes, the remaining hands are not all Ace-high flushes, because there are five card straight flushes that utilize the joker. It's possible to calculate the combinations for straight flushes utilizing a joker, but I don't have the time or energy to do that right now.

I recently got a 6-card Straight Flush in UTH. This is the highest achievement in my whole life. Let me estimate the odds as follows:

The odds of a 5-card Straight Flush is 3,590-to-1. Multiply this number by 47/2, I get an odds of 3590x47/2= 84365, which is almost 3 times rarer than a Royal Flush. Correct?

I calculate that the odds getting a 6 card straight flush in 7 cards is P= (7*4*44+ 2*4*45)/combin(52,7) = 1.18997E-05 which is 1 in 84035.52764.

Quote: gordonm888

I calculate that the odds getting a 6 card straight flush in 7 cards is P= (7*4*44+ 2*4*45)/combin(52,7) = 1.18997E-05 which is 1 in 84035.52764.
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So about once in 84,000 hands. Now I have some research to do as to hands per day. Thanks, this may be interesting.