Respectfully ask for help with the following. 2400 Slanksky bucks sent electronically to anyone that can help.

House Banked NLH Side Bet.

Based only on Flop. Player bets a $1 against the house pre flop (dollar amount of bet doesnt really matter though)

Does not have anything to do with players hole cards. All they do is pick in advance three cards (suits dont matter). So they pick 10, J ,8, or A,4,9 etc.

Then flop comes and bet is paid according to following pay table

If Flop comes any 3 of a Kind. Pays 100:1

If Flop comes any combination of their three cards. Pays 75:1

If Flop comes any flush. Pays 10:1

Any pair on flop. Pays 1.5: 1

Question.

What is theoretical house edge (hold?) on this bet?

How could above pay table be adjusted so that house edge is ~10% ?

Giving it a try, not sure if I got correctly all the cases. I'll give you only the probabilities of each winning case, leaving to you the final tally.

Say that your pick is A23. I also assume that you make the pick before seeing your hole cards. Also, I'm gonna neglect any poker consideration. It is well known, for instance, that an Ace appears on the flop slightly less than its "uniform" share, since people tend to get involved in a hand and see flops with high cards.

Giving these assumptions, you have C(52,3) = 22100 possible flops.

Guessing that here you win here with also, say, 444 and not only with AAA/222/333. If so, you win 13 * 4 times (13 ranks and for each rank you have 4 different 3 of a kind). So, the probability is:

13*4/22100 = 0.0023529411764706


So, you win with A23, but also with AA2 or similar. We need to remove AAA/222/333 because case #1. We got:

(C(12,3)-3*4) / 22100 = 0.0094117647058824

We need here to remove A23 suited, because case #2.

(4*C(13,3)-4) / 22100 = 0.05158371040724


We need to remove AA2 flop types because case #2.

(6*13*4*12 - 6*3*4*2) / 22100 = 0.16289592760181

I might have made mistakes in counting.

Nick.

awesome thanks for getting this framed up. lemm see if can take your above work and answer my question.

Looking at event frequency and pay table, am I correct in the following?

On a $1 bet....


3 of Kind hits 0.0023 Pays $100 so EV is $.23

Combo of 3 Cards hits 0.0094 Pays $75 so EV is $.70

Flush hits 0.0515 Pays $10 so EV is $.51

Any Pair hits 0.162 Pays $1.5 so EV ia $.24


Thus Total EV of $1 Bet is $1.70 for the player. Did I use your math correctly?

If so, sounds like I need to adjust the pay table Casinos weren't build by offering bets where players have an edge.

You need to subtract from that all the losing bets. You lose a dollar .7748 of the time.

1.70 - .77 = negative EV of .07.

ah quite true. I'm a noob. very much appreciate helping me understand this better.

So adjusting my math based on what you said.

Side bet math looks like

Prob Pay EV
--------------------------------------------------------------

3 of Kind 0.2% 100 $.24

any comb of 3 cards 0.9% 75 $.71

any pair 16.3% 1.5 $.24

Losing Bet 77.4% -1 -$.77
---------------------------------------------------------------
Payout to Player $.93
Hold/Edge for House $.07

(sorry cant make table format with tabs or spaces)


So.... Final Answer. House Edge on bet described in OP is 7%

Is this correct ?

You left out Flush. Also, house edge is not "hold". That is something different entirely, and in most games it will be a lot higher the the edge due to betting repeatedly with the same money.

woops. had right in excel so totals are right. here is correct table

good point on edge/hold. I still use those incorrectly.

Since there is no skill in this bet (compared to Black Jack) the house edge will be exactly above. Where in BJ edge is higher than theoretical due to imperfect play. Is that a correct statement?

Hold would be total dollar in box at end of night . A function of edge PLUS number of bets placed? is that a correct statement?

Casino sells you $100 in chips. You bet $1 at a time on a game where the house has a 1% edge. You bet 1000 times, losing an average of .01 per bet, so you lose $10. You cash in your chips for $90. The house hold is $10 of your original $100, so 10%.

So in this example edge is 1%. Hold is 10%.

Hold can't really be calculated in advance on a new game, only measured, because it depends on player behavior.

got it.

really appreciate the help.

PMing you and Nick 1200 Slansky Bucks each. Dont spend them all in once place