rjs357
• Posts: 12
Joined: Jun 29, 2014
July 8th, 2014 at 2:00:59 PM permalink
The number 19,933,230,517,200 comes up as the total number of combinations in Jacks or Better Video Poker.
Would somebody please provide a formula for this number in terms of combinations like C(52,5) (which is 52!/47!5!)?
Please include underlying hypothesis such as number of decks or whatever. The number is the size of the complete
sample space of something. The number is so large I can't guess what that sample space is. Thanks and I'm very
grateful for your help. Also, if you could suggest a good book or a website on probability calculations underlying card games
like Five Card Draw, that would be great. Thanks again.
Wizard
• Posts: 26842
Joined: Oct 14, 2009
July 8th, 2014 at 2:24:14 PM permalink
Quote: rjs357

The number 19,933,230,517,200 comes up as the total number of combinations in Jacks or Better Video Poker....

This is a little hard to explain, so read carefully.

First, there are C(52,5)=2,598,960 ways to deal the initial five cards.

The number of possible combinations on the draw depends on the number of cards the player discards. The following table shows the draw combinations in the second column.

0 1 7,669,695 7,669,695
1 47 163,185 7,669,695
2 1,081 7,095 7,669,695
3 16,215 473 7,669,695
4 178,365 43 7,669,695
5 1,533,939 5 7,669,695

However, the weight of hands on the draw need to be weighted according to how many cards the player discarded. For example, a royal after keeping four to a royal is much more likely than a royal after tossing everything. To keep the combinations integers we need to find the least common multiple of the draw combinations. The answer to that is 7,669,695, which equals 5*C(47,5)

So C(52,5)*C(47,5)*5 = 19,933,230,517,200.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
AxiomOfChoice
• Posts: 5761
Joined: Sep 12, 2012
July 8th, 2014 at 2:26:22 PM permalink
Quote: rjs357

The number 19,933,230,517,200 comes up as the total number of combinations in Jacks or Better Video Poker.
Would somebody please provide a formula for this number in terms of combinations like C(52,5) (which is 52!/47!5!)?
Please include underlying hypothesis such as number of decks or whatever. The number is the size of the complete
sample space of something. The number is so large I can't guess what that sample space is. Thanks and I'm very
grateful for your help. Also, if you could suggest a good book or a website on probability calculations underlying card games
like Five Card Draw, that would be great. Thanks again.

What do you mean, it comes up? Where did you get it from?

At first I thought that it might be the total number of possible (deal, draw) pairs but it looks too big even for that (it's almost triple the the square of the number of deals)
Ahigh
• Posts: 5198
Joined: May 19, 2010
July 8th, 2014 at 2:55:09 PM permalink
One question related to this has to do with least optimal play. It's well known that video poker offers advantage play, and I often wonder what actual observed hold percentages are for some machines. But the absolute lowest payback percentage difference based on intentionally holding the worst possible cards every time is a number that I am not familiar with nor have I ever heard of anyone determining (least optimal play backback percentage based on a given game and pay table).
aahigh.com
tringlomane
• Posts: 6282
Joined: Aug 25, 2012
July 8th, 2014 at 2:56:01 PM permalink
Quote: rjs357

Also, if you could suggest a good book or a website on probability calculations underlying card games
like Five Card Draw, that would be great. Thanks again.

For actual 5 card draw, this book shows calculations in significant detail.

http://www.amazon.com/DRAW-POKER-ODDS-Mathematics-Classical/dp/9738752051

For video poker, Stuart Ethier offers an analysis of "video poker" as the sample chapter of his book "The Doctrine of Chances" on his faculty website.

http://www.math.utah.edu/~ethier/sample.pdf
AxiomOfChoice
• Posts: 5761
Joined: Sep 12, 2012
July 8th, 2014 at 3:00:11 PM permalink
Quote: AxiomOfChoice

What do you mean, it comes up? Where did you get it from?

At first I thought that it might be the total number of possible (deal, draw) pairs but it looks too big even for that (it's almost triple the the square of the number of deals)

Oh, I somehow missed the wizard's reply before I wrote mine. Never mind..

(edit: I actually think that we were writing at the same time, so his reply was not there before I started replying but was there by the time I hit "post")
tringlomane
• Posts: 6282
Joined: Aug 25, 2012
July 8th, 2014 at 3:17:39 PM permalink
Quote: Ahigh

One question related to this has to do with least optimal play. It's well known that video poker offers advantage play, and I often wonder what actual observed hold percentages are for some machines. But the absolute lowest payback percentage difference based on intentionally holding the worst possible cards every time is a number that I am not familiar with nor have I ever heard of anyone determining (least optimal play backback percentage based on a given game and pay table).

We did it awhile back for 9/6 JoB (99.54% return w/optimal strategy). It's a 2.3717% return if you are allowed to throw away a dealt Royal. Most machines don't let you do that, so the return one would realistically expect is 2.495%.

https://wizardofvegas.com/forum/gambling/video-poker/14822-theoretical-payout-on-9-6-job-with-opposite-to-optimal-strategy/

As for actual "return", I've read that it's often 2 to 4% below optimal. It depends on the game and market though. JoB is played much more accurately than Ultimate X for example. I would also expect players on the Strip to play video poker worse because they are more often trying the game for the first time than some Vegas local.
AxelWolf
• Posts: 22317
Joined: Oct 10, 2012
July 9th, 2014 at 3:35:46 AM permalink
Quote: tringlomane

We did it awhile back for 9/6 JoB (99.54% return w/optimal strategy). It's a 2.3717% return if you are allowed to throw away a dealt Royal. Most machines don't let you do that, so the return one would realistically expect is 2.495%.

https://wizardofvegas.com/forum/gambling/video-poker/14822-theoretical-payout-on-9-6-job-with-opposite-to-optimal-strategy/

As for actual "return", I've read that it's often 2 to 4% below optimal. It depends on the game and market though. JoB is played much more accurately than Ultimate X for example. I would also expect players on the Strip to play video poker worse because they are more often trying the game for the first time than some Vegas local.

so your saying if you make the worst possible holds you would only get back 2.3%. I thought it would be higher then that. Sometimes no matter what you do you end up with a pay.
♪♪Now you swear and kick and beg us That you're not a gamblin' man Then you find you're back in Vegas With a handle in your hand♪♪ Your black cards can make you money So you hide them when you're able In the land of casinos and money You must put them on the table♪♪ You go back Jack do it again roulette wheels turinin' 'round and 'round♪♪ You go back Jack do it again♪♪
JB
• Posts: 2089
Joined: Oct 14, 2009
July 9th, 2014 at 4:53:07 AM permalink
Quote: AxelWolf

so your saying if you make the worst possible holds you would only get back 2.3%. I thought it would be higher then that. Sometimes no matter what you do you end up with a pay.

I think you're overlooking the multitude of losing deal hands, such as 2-3-6-10-Q, where you would hold everything.
dwheatley
• Posts: 1246
Joined: Nov 16, 2009
July 9th, 2014 at 6:09:03 AM permalink
Quote: AxiomOfChoice

Oh, I somehow missed the wizard's reply before I wrote mine. Never mind..

(edit: I actually think that we were writing at the same time, so his reply was not there before I started replying but was there by the time I hit "post")

Ninja'd. Makes us all look silly one time or another.
Wisdom is the quality that keeps you out of situations where you would otherwise need it
AxiomOfChoice
• Posts: 5761
Joined: Sep 12, 2012
July 9th, 2014 at 12:07:45 PM permalink
Quote: dwheatley

Ninja'd. Makes us all look silly one time or another.

Except that the Wizard's answer was the right one and mine was just asking wtf was going on :)
rjs357
• Posts: 12
Joined: Jun 29, 2014
July 11th, 2014 at 1:01:02 PM permalink
I understand how you arrived at 19,933,230,517,200.
I see that 5*C(47,5) is the LCM of the set {C(47,n)| n=0,1,2,3,4,5} but I'm struggling to find a probabilistic justification to your approach,
"To keep the combinations integers we need to find the least common multiple of the draw combinations."

I want to respectfully submit the following alternative.
The video poker player starts with 5 cards, there are C(52,5) possibilities.
From there, Player draws n cards, where n = 0,1,2,3,4,5.
Player must choose n cards from his/her 5 card hand, there are C(5,n) ways of doing this.
Player is then dealt n cards from the remaining 47 card deck, there are C(47,n) possible outcomes to the draw.
From the sum rule for probabilities of nonintersecting events, summing over n should lead to the size of our sample space,
C(52,5)[C(5,0)*C(47,0) + C(5,1)*C(47,1) + C(5,2)*C(47,2) + C(5,3)*C(47,3) + C(5,4)*C(47,4) + C(5,5)*C(47,5)]
= 2,598,960 * 2,598,960 = C(52,5)^2.
The formula for the factor with the sum can be obtained by direct calculation of from Vandermonde's Identity.
This is the value mentioned by AxiomOfChoice in the post following the your post.
As was pointed out there by AxiomOfChoice, the ratio 19,933,230,517,200 / C(52,5)^2 is about 2.95.

Would you please explain in just a little more detail how your LCM approach relates to the size of our sample space?

Thanks
rjs357