Poker Calculations: Video Poker

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fira United States. Dec 06 2015 01:49. Posts 6345

((Just some random poker calculations I did via brute force. I'd appreciate it if someone more knowledgeable could confirm the legitimacy of these calculations, whether it be the results or the "methods"!))

Game: Video Poker Jacks or Better

Payout chart:
~Royal flush 250x
~Straight flush 50x
~4OAK 25x
~Full House 9x
~Flush 6x
~Straight 4x
~3OAK 3x
~2Pair 2x
~JJ+ 1x

Our Hand: Ad Jc Xx Yy Zz

What do we do? Options:
~hold Ad
~hold Ad, Jc

========================================================================================================================
Option 1: hold Ad:
if we hold just Ad, we discard the other four cards and draw 4 new cards
our total EV should equal:
(chance of getting royal flush * 250) +
(chance of getting straight flush * 50) +
(chance of getting 4OAK * 25) +
(chance of getting full house * 9) +
(chance of getting flush * 6) +
(chance of getting straight * 4) +
(chance of getting 3OAK * 3) +
(chance of getting 2Pair * 2) +
(chance of getting JJ+ * 1)

chance of getting royal flush = 4/50 * 3/49 * 2/48 * 1/47 = 0.00000434216 = 1/230,300
chance of getting straight flush = 4/50 * 3/49 * 2/48 * 1/47 = 0.00000434216 = 1/230,300

chance of getting 4OAK =
--4 ways to get four Aces--
{A, A, A, X}
{A, A, X, A}
{A, X, A, A}
{X, A, A, A} (3/50 * 2/49 * 1/48) = 5.10204082E-5 = 1/19,600
--1 way to get four Xs--
{X, X, X, X} (47/50 * 3/49 * 2/48 * 1/47) = 5.10204082E-5 = 1/19,600
Total = 5 * 5.10204082E-5 = 0.000255102041 = 1/3,920

chance of getting full house =
--6 ways to get Aces full--
{A, A, X, X}
{A, X, A, X}
{A, X, X, A}
{X, A, A, X}
{X, A, X, A}
{X, X, A, A} (47/50 * 3/49 * 3/48 * 2/47) = 0.00015306122 = 1/6,533
--4 ways to get X's full--
{A, X, X, X}
{X, A, X, X}
{X, X, A, X}
{X, X, X, A} (47/50 * 3/49 * 2/48 * 3/47) = 0.00015306122 = 1/6,533
Total = 10 * 0.00015306122 = 0.0015306122 = 1/653

chance of getting flush =
{Wd, Xd, Yd, Zd} (12/50 * 11/49 * 10/48 * 9/47) = 0.00214937038 = 1/465

chance of getting straight =
--2 ways to get straight--
{2,3,4,5} (16/50 * 12/49 * 8/48 * 4/47) = 0.00111159357 = 1/899
{KQT,KQT,KQT,J} (12/50 * 8/49 * 4/48 * 3/47) = 0.00020842379 = 1/4798
{KQT,KQT,J,KQT}
{KQT,J,KQT,KQT}
{J,KQT,KQT,KQT}
Total = 0.00111159357 + (4 * 0.00020842379) = 0.00194528873 = 1/514

chance of getting 3OAK =
--6 ways to get trip Aces--
{A, A, X, Y}
{A, X, A, Y}
{A, X, Y, A}
{X, A, A, Y}
{X, A, Y, A}
{X, Y, A, A} (47/50 * 43/49 * 3/48 * 2/47) = 0.00219387755 = 1/456
--4 ways to get trip Xs--
{X, X, X, Y}
{X, X, Y, X}
{X, Y, X, X}
{Y, X, X, X} (47/50 * 43/49 * 3/48 * 2/47) = 0.00219387755 = 1/456
Total = 10 * 0.00219387755 = 0.0219387755 = 1/46

chance of getting 2Pair =
--10 ways to get Aces up--
{A, X, X, Y}
{A, X, Y, X}
{A, Y, X, X}
{X, A, X, Y}
{Y, A, X, X}
{X, X, A, Y}
{X, Y, A, X}
{X, X, Y, A}
{X, Y, X, A}
{Y, X, X, A} (47/50 * 43/49 * 3/48 * 3/47) = 0.00329081632 = 1/304
--6 ways to get other 2Pair--
{X, X, Y, Y}
{X, Y, X, Y}
{X, Y, Y, X}
{Y, X, X, Y}
{Y, X, Y, X}
{Y, Y, X, X} (47/50 * 3/49 * 43/48 * 3/47) = 0.00329081632 = 1/304
Total = 16 * 0.00329081632 = 0.05265306112 = 1/19

chance of getting JJ+ =
--4 ways to get pair of Aces--
{A, X, Y, Z}
{X, A, Y, Z}
{X, Y, A, Z}
{X, Y, Z, A} = (47/50 * 43/49 * 39/48 * 3/47) = 0.04278061224 = 1/23
--6 ways to get pair of Kings--
{K, K, X, Y}
{K, X, K, Y}
{K, X, Y, K}
{X, K, K, Y}
{X, K, Y, K}
{X, Y, K, K} = (43/50 * 40/49 * 4/48 * 3/47) = 0.00373425966 = 1/268
--6 ways to get pair of Queens--
same as Kings
--6 ways to get pair of Jacks--
{X, Y, J, J} = (43/50 * 40/49 * 3/48 * 2/47) = 0.00186712983 = 1/536
Total = (0.04278061224 * 4) + (0.00373425966 * 12) + (0.00186712983 * 6) = 0.22713634386 = 1/4.4

Total EV of keeping just Ad =
(0.00000434216 * 250) +
(0.00000434216 * 50) +
(0.00025835866 * 25) +
(0.0015306122 * 9) +
(0.00214937038 * 6) +
(0.00194528873 * 4) +
(0.0219387755 * 3) +
(0.05265306112 * 2) +
(0.22713634386 * 1) = 0.4404732941
========================================================================================================================
========================================================================================================================
Option 2 - hold Ad, Jc:
if we hold both Ad and Jc, we discard the other three cards and draw 3 new cards
our total EV should equal:
(chance of getting royal flush * 250) +
(chance of getting straight flush * 50) +
(chance of getting 4OAK * 25) +
(chance of getting full house * 9) +
(chance of getting flush * 6) +
(chance of getting straight * 4) +
(chance of getting 3OAK * 3) +
(chance of getting 2Pair * 2) +
(chance of getting JJ+ * 1)

chance of getting royal flush = 0
chance of getting straight flush = 0
chance of getting 4OAK = (3/50 * 2/49 * 1/48) * 2 = 0.00010204081 = 1/9800

chance of getting full house =
--3 ways to get Aces full of Jacks--
{A, A, J}
{A, J, A}
{J, A, A}
--3 ways to get Jacks full of Aces--
{A, J, J}
{J, A, J}
{J, J, A} = (3/50 * 2/49 * 3/48) = 0.00015306122 = 1/6,533
Total = 6 * 0.00015306122 = 0.00091836732 = 1/1089

chance of getting flush = 0

chance of getting straight =
{KQT, KQT, KQT} (12/50 * 8/49 * 4/48) = 0.00326530612 = 1/306

chance of getting 3OAK =
--3 ways to get trip Aces--
{A, A, X}
{A, X, A}
{X, A, A}
--3 ways to get trip Jacks--
{J, J, X}
{J, X, J}
{X, J, J} = (44/50 * 3/49 * 2/48) = 0.00224489795 = 1/445
--1 way to get trip Xs--
{X, X, X} = (44/50 * 3/49 * 2/48)
Total = 7 * 0.00224489795 = 0.0157142857 = 1/64

chance of getting 2Pair =
--3 ways to get Aces up (non-Jacks)--
{A, X, X}
{X, A, X}
{X, X, A} = (44/50 * 3/49 * 3/48) = 0.00336734693 = 1/297
--3 ways to get Jacks up (non-Aces)--
{J, X, X}
{X, J, X}
{X, X, J}
--6 ways to get Aces and Jacks--
{A, J, X}
{A, X, J}
{J, A, X}
{J, X, A}
{X, A, J}
{X, J, A} = (44/50 * 3/49 * 3/48) = 0.00336734693 = 1/297
Total = 12 * 0.00336734693 = 0.04040816316 = 1/25

chance of getting JJ+ =
--3 ways to get pair of Aces--
{A, X, Y}
{X, A, Y}
{X, Y, A} = (44/50 * 40/49 * 3/48) = 0.04489795918 = 1/22
--3 ways to get pair of Jacks--
{J, X, Y}
{X, J, Y}
{X, Y, J}
--3 ways to get pair of Kings--
{K, K, X}
{K, X, K}
{X, K, K} = (40/50 * 4/49 * 3/48) = 0.00408163265 = 1/245
--3 ways to get pair of Queens--
{Q, Q, X}
{Q, X, Q}
{X, Q, Q}
Total = (0.04489795918 * 6) + (0.00408163265 * 6) = 0.29387755098 = 1/3.4

Total EV of keeping Ad and Jc =
(0 * 250) +
(0 * 50) +
(0.00010204081 * 25) +
(0.00091836732 * 9) +
(0 * 6) +
(0.00326530612 * 4) +
(0.0157142857 * 3) +
(0.04040816316 * 2) +
(0.29387755098 * 1) = 0.44571428501
========================================================================================================================

Comparison--------------Hold Ad-------------Hold Ad, Jc--------
Royal flush EV----------0.00108554----------0------------------
Straight flush EV-------0.000217108---------0------------------
4OAK EV-----------------0.0064589665--------0.00255102025------
Full house EV-----------0.0137755098--------0.00826530588------
Flush EV----------------0.01289622228-------0------------------
Straight EV-------------0.00778115492-------0.01306122448------
3OAK EV-----------------0.0658163265--------0.0471428571-------
2Pair EV----------------0.10530612224-------0.08081632632------
JJ+ EV------------------0.22713634386-------0.29387755098------
Total EV----------------0.4404732941--------0.44571428486------

0 votes

devon06atX Canada. Dec 06 2015 05:25. Posts 5459

So. Fira. What's up?

K40Cheddar United States. Dec 07 2015 17:35. Posts 2202

www.wizardofodds.com

fira United States. Dec 07 2015 22:34. Posts 6345

On December 06 2015 04:25 devon06atX wrote:
So. Fira. What's up?

friend wanted me to post a message here... dunno why he doesn't just post it himself! but anyway:

Sup Chen. Hi Devon. You probably don't recall me Devon. We played a lot of hands at NL25. I was ThePatient9. You were positive against me, LOL, although not against the limit?

Video poker is a pretty cool game, from what I can tell. I played a $50R on a cruise and placed third after spending $200. I won ~$900 from third place when the young guy sucked out AJ vs my AQ high (flop AI). That's the turn for you, LOL, never lucky Thai dealers, heh.

A black marine and his white marine wife won $35,000 from a Royal Flush on the same cruise. Bet they wish they had wagered more than five bucks, LOL!

Last edit: 07/12/2015 22:34

failsafe United States. Dec 08 2015 01:09. Posts 1040

Thx for posting chenaments, couldn't access site all day

devon06atX Canada. Dec 10 2015 10:30. Posts 5459

On December 07 2015 21:34 fira wrote:

Show nested quote +
On December 06 2015 04:25 devon06atX wrote:
So. Fira. What's up?

friend wanted me to post a message here... dunno why he doesn't just post it himself! but anyway:

Sup Chen. Hi Devon. You probably don't recall me Devon. We played a lot of hands at NL25. I was ThePatient9. You were positive against me, LOL, although not against the limit?

Video poker is a pretty cool game, from what I can tell. I played a $50R on a cruise and placed third after spending $200. I won ~$900 from third place when the young guy sucked out AJ vs my AQ high (flop AI). That's the turn for you, LOL, never lucky Thai dealers, heh.

A black marine and his white marine wife won $35,000 from a Royal Flush on the same cruise. Bet they wish they had wagered more than five bucks, LOL!

Nah, no offense - don't recall you dude. Sure you're a solid player and all.

And yep. Basically a big loser at every stake I play. Luckily, I have Liquidpokers Fine Rakeback Offers at the most competitive sites to keep me afloat.

failsafe United States. Dec 14 2015 23:46. Posts 1040

Can't believe you don't remember me. My play is mathematically perfect. I learned from the legendary vital myth

failsafe United States. Dec 14 2015 23:52. Posts 1040

Honestly I think I'm the best player of all time. After running something like 20bb/100 below expectation I finished 10bb/100 ahead over 10,000 hand sample at NL50 FTP 6tabling on speed. Obviously speed dramatically lowers your luck, but I compensated with perfect TAG play.

failsafe United States. Dec 14 2015 23:54. Posts 1040

The worst experience of my life was playing limit Stars 1/2 at a friend's house. I get AK hit the ace on flop max out FTR lose to AQ turn Q, and the guy who the account belonged to actually had the gall to ridicule my play. Fucking limit players are seriously bad.

fira United States. Dec 15 2015 03:00. Posts 6345

lol limit

lol playing online poker at other peoples houses

AstroMech becomes 150% troll whenever hes at my house obsing me... basically i have to take the opposite of whatever he says

failsafe United States. Dec 15 2015 08:06. Posts 1040

I GOT SOME PCP LACED SATIVA AND VYVANSE AND PLAYED NL100 RAPED THE LIMIT WHILE SCOTT LAYED ON MY BED. BULLYING SHOVING CRUSHING FULL CALEB STYLE THEN QQ ALL-IN INTO KKPF FUCKING CRUSHED LOST $200 ONE TABLING OVER 150 HANDS WITH PERFECT PLAY FTRIGED

failsafe United States. Dec 18 2015 06:32. Posts 1040

HELLO FIRA. I AM HERE TO ANNOUNCE THE COMING OF THE LORD. WELCOME TO THE HOUSE OF THE LORD. HOUSE OF THE LORD. JESUS CHRIST SPEAKING.

fira United States. Jan 08 2016 09:31. Posts 6345

LOL HAI WADDUP JSUS