NLHE Side Bet: How Often Do Winning Hands Include a J

Mr. Wizard,

First off, love the site and all the neat little math facts and questions you have answered. I have a silly one, but here it goes...

Interesting side bet during cash games came up the other day.

A gentleman offered me a 1:1 side bet in a NLHE ring game, 9 handed.

The bet was that a Jack would be part of the winning hand at showdown. I thought that was gonna happen less than 50% of the time so I accepted, however the gentlemen insisted that I had to agree to make the bet over the course of at least 20 hands (at $10 each hand). That shouldn't have mattered and I had extra funds for the bet so I accepted.

It took awhile for the bet to be resolved to completion as we where only counting hands that made it to showdown, and not hands that where shown as bluffs.

The gentleman ended up winning the bet overall, with 13 out of 20 winning show down hands containing a J (either because there was a J on the board that was counted or because the player had a J in their pockets that played.

So, I was wondering was the guy lucky or is there some sorta mathematical quirk in NLHE that gave him the edge?

Thanks!

DJ

I don't know the numbers, but ask yourself this question: How often would the winning hand contain a 4?

Letme see, 52 cards, 9 cards revealed at showdown (at least) so 9/52 is 17.3% of any one card, regardless of whether that card is part of a winning hand (I think).

Each player extra in the pot adds +3.8% to the chances?

But the guy won 13/20 which 65%?!?

I dunno, maybe. But I was just wondering if it has to do with the way hand ranges work in poker (presuming you are even playing with people that care)?

Like, are Jacks specifically way more apt to be played much more frequently than other cards like, say, a 4?

I know in my personal hand range it is more likely than most other cards, but I have less Jacks than I do Aces, Kings, and Queens, which is duh(!) because those cards are stronger than Jacks.

And, most of the time (going from memory here) it seemed the Jack was just part of the winning hand by virtue of being a community card (one player has a single pair, a Jack in the community run out almost always plays).

The bet specifically excluded hands that didnt make it to showdown, not sure if that mattered.

Anyway, maybe the guy just got really lucky. Winning 6.5/10 with naked odds of 1.7/10 isnt the craziest thing Ive seen (saw a guy run it twice with only 2 outs catch his outs *both* times once which was absolutely mindblowing so this aint nothing to that!!!)

Anyway, I have no idea how to break it down more than the basic odds that a single card will appear, which is why I figured I'd ask on this site. :)

1 - [(C(4,0)*C(48,9)) / C(52,9)]

easy to use a calculator

wolframalpha

simply
1 - P(no Jacks)

4209 / 7735

54.41%

[(C(4,0)*C(48,9)) / C(52,9)]

4 Jacks Choose0 * 48 Others Choose 9 / 52 total cards Choose 9

distribution for # of Jacks
more math involved ( 4 those that enjoy it)
# Jacks / Prob
0/ 0.455850032
1/ 0.410265029
2/ 0.120077569
3/ 0.013341952
4/ 0.000465417

Excel in Google
https://goo.gl/Tz55kF

fun question
Sally

looks like you were not favored

Quote: DJVILLAIN

Anyway, maybe the guy just got really lucky.

4209 / 7735 = probability of winning one time
this is binomial probability

another calculator
http://vassarstats.net/binomialX.html

parameters
n=20
k=13
p=4209 / 7735

P: 13 or more out of 20
0.235438939735

one can also use the normal distribution too

Sally

Quote: DJTeddyBear

I don't know the numbers, but ask yourself this question: How often would the winning hand contain a 4?

Phil Hellmuth, almost never.

Gus Hansen, almost a lot.

Wow, thanks Sally!

My calculator doesn't have all those letters in it, though! ;P

Yikes, so instead of me being hugely favored it was actually pretty close to a coin flip slightly weighted in his favor!

Drats, well next time I will insist on getting paid 2:1 odds or something like that.

No wonder I am such a fish.

Thanks!

Lolz Very True!

So if I am at a table full of nits then it is a bad bet, but if I am playing at a table full of LAGs then its a good bet!

Quote: DJVILLAIN

...Yikes, so instead of me being hugely favored it was actually pretty close to a coin flip slightly weighted in his favor!...

How did you think you were ever hugely favored? Even in your quick analysis you saw how difficult it is to get 1 specific card on the board or in the hands... 9/52.

I would have figured you're a huge dog from the beginning. Now past the "mathematics" and more to the "play" of the game, it's well known that poker players often play the bigger more "premium" cards/hands. Thus, I would have asked him for an Ace instead of a Jack. While the math might be similar, it fails to take in to account that poker players will absolutely more likely play an ace with a random card than a Jack with a random card.

So earlier when someone said what's the difference between a Jack and a 4? Well, mathematically not much, but in the game of poker a lot (in my opinion). People play the higher cards. Say they got a King as their other card... 4-K folds where as J-K plays, for example.

So while the math might not be in your favor, if you could get 2 cards or an Ace I would think this would be a good bet. Straight off the top though my gut tells me just a Jack is a really bad bet for you.

Well, yeah. I was betting *against* a jack being part of the winning hands at showdown for all the reasons you listed.

Thing is, the bloody jacks kept making appearances.

I *thought* I was hugely favored because I was thinking there was only a 1 in 6 or so of a jack showing up (my math was very incorrect) and being part of the winning hand.

If he had said A or K maybe I wouldnt have taken the bet because, ya know, 9-handed its pretty normal to expect winning hands to include more of those.

Anyways, who would have thought that such a high percentage of winning hands would be associated with a J in some way?

Can you clarify that if the J was on the board, that it had to be part of the five card hand or not?

Examples:

1) Board: J5522 rainbow. Player: A,Q offsuit. Hand is A5522. Here the J doesn't play.

2) Board: J9742 rainbow. Player: A,Q offsuit. Hand is AQJ97. Here the J does play.

The J had to be part of the winning hand, at showdown on the river. Hands that had no showdown or did not make it to the river were disregarded even if someone showed a bluff that included a Jack.

Hands that got multiple run outs counted once per run out (only came up once and we had to clarify the bet on that issue and I was a dumass and agreed to count each run because it was a made Q high straight vs flush draw and I was hoping to get lucky but anyways).

Ah Kh Jh 6d 6c Winner: Ac 6h
Jack doesnt play, so bet loses.

Same board, Winner: 8h 7h
Jack plays, so bet wins.

Community: 6h 3c 8d 9s Ah Winner: Jh 9c
Jack plays, bet wins.

Community: Kc 9s 7d Jh 3c Winner: As 5s
Jack plays, bet wins

Most of the time if a J was in the community cards it got counted when the winner made only pairs, two pairs, sets and trips. Or ace high, of course.

So almost everytime a J hit the community cards it played, and when it didnt I still had to sweat run outs that included broadway draws, flush draws, and medium str8 draws lol.

Thanks.

Actually, even though my expectations were way off on the odds of a Jack being in a winning hand, after coming here and seeing the math I dont feel so bad.

In order to make the J part of the winning hand you only use the *winners* hole cards plus the five community cards - unless I am mistaken Sally's excellent example of mathematics in motion counted all 9 cards (including the losing player who doesn't count).

When I replaced the 9 with a 7 in the nifty online calculator she linked me to I got only a 44% chance of a J (and that just calcs that a J will be in those 7 cards, so it still might not play).

Course I could be doing it wrong, I suppose, and, yes, player ranges would heavily factor into whether the J counts; that being said it really looks like its close to a coin flip bet, and the guy got lucky to win 13 times.

So, I may still be a fish but I wasnt a huge sucker this time. xD

Quote: DJVILLAIN

...So, I may still be a fish but I wasnt a huge sucker this time. xD

Ah, so you were betting AGAINST the J... and it couldn't be in the showdown hands it had to be IN the 5 card winning hand. That's interesting.

I'm curious now too what the final numbers are, but admittedly at face value, I'd bet against that. Remember though no online calculator will take in to effect poker players choosing to play better cards... so if the raw numbers come out to be like 54-46 in your favor, remember the weighted factor of players playing "face cards" more often, which might level it out or tip it the other way even... hard to quantify.

Quote: DJVILLAIN

Course I could be doing it wrong, I suppose,

I did not understand the question at first read, so I did answer a different question. Have to look at it again.
still a fun question
without taking into account if a J would be played
Sally

You are awesome Sally, thanks!

Ya know, I worked with money for 20 years almost and never cared a like for math beyond basic adding, subtracting, multiplying and dividing. Never needed it.

Then *this guy* decided he loved poker and I think I am just so far behind. lolz.