Break even for heads up holdem progressive

Hi does anyone know the break even point for heads up holdem progressive?

I would need all kinds of information:

1.) Main Game or Side Bet

2.) If Main Game, what is the paytable and what triggers the Progressive?

3.) If Side Bet, """""""""""""""""""""""""""""""""""""""""""""""""""""""""""""'

Thanks for quick reply it is a side bet,you must have two royal cards in your hand
Pay table for side bet is
Jackpot must have two royal cards in hand
1000:1 royal flush using community cards
250:1 straight flush
75:1 four of a kind
11:1 full house
Main game payouts are
500:1 royal flush
50:1 straight flush
25:1 four of a kind
3:1 full house
1.5 to 1 flush
1:1 full house

Quote: nj2741

Thanks for quick reply it is a side bet,you must have two royal cards in your hand
Pay table for side bet is
Jackpot must have two royal cards in hand
1000:1 royal flush using community cards
250:1 straight flush
75:1 four of a kind
11:1 full house

No problem, I'm only going to worry about the side bet for these purposes, so let's see what it takes to make that positive.

This seems similar to the, "Small Progressive," that can be found on this page:

https://wizardofodds.com/games/ultimate-texas-hold-em/

The main difference is that the Full House pays one more unit and the Progressive that you are referring to requires you to use both of the cards in your hand, as opposed to just one, to hit the Progressive. Otherwise, the probabilities are going to be the same for the other hands.

Probability of Full House: 0.025961 Return on FH: .025961 * 11 = .285571
Probability of 4OaK: 0.001681 Return on 4OaK: .001681 * 75 = 0.126050
Probability of SF: 0.000279 Return on SF: .000279 * 250 = .069627
Probability of Loss: -0.972047

Given those things, we know that the expected return without the Royal is: (.285571 + .126050 + .06927 - .972047) = -0.491156

That leads us to this very important question: Is there an Envy Bonus, and if so, how much does it pay? Once I know that, I can finish the rest.

Nope no envy bonus

No Envy Bonus is sweet! Not for you, of course, but it prevents me from having to consider players at the table and all of that.

Okay, let's go ahead and carry down what we started with above:

Probability of Full House: 0.025961 Return on FH: .025961 * 11 = .285571
Probability of 4OaK: 0.001681 Return on 4OaK: .001681 * 75 = 0.126050
Probability of SF: 0.000279 Return on SF: .000279 * 250 = .069627
Probability of Loss: -0.972047

Given those things, we know that the expected return without the Royal is: (.285571 + .126050 + .06927 - .972047) = -0.491156

The first thing that we know is that the probability of any Royal Flush, given seven cards, is 0.0032%, or 1/.000032 = 1 in 31,250

Given that, we must isolate Royals in which we use both of our cards from Royals in which we do not. That's actually a simpler matter than it would seem. The reason why is we already know there is a total of seven cards, so we ask ourselves, of those seven, what is the probability that we got two of them in our starting hand.

(5/7) * (4/6) = 0.47619047619

That means that 47.619047619% of all of those Royals will see us using both of our cards. Just to check it out and verify this hypothesis, let us first look at the possibility of neither of our cards being Royal cards. (All community)

(2/7) * (1/6) = .04761904761

That means that 4.761904761% of all hands in which we do get a Royal will have us seeing neither of the five Royal cards in our hand. That too can be somewhat verified:

.04761904761 * .000032 = .0000015238 or 1 in 656254.101588

We note that those are roughly the Odds of drawing a Royal using five cards. There are going to be some differences due to rounding as well as the fact that there are two cards out.

That takes us to our next scenario, which is to have one card match a Royal card but then to have another card not do that. Here is how that plays out:

(2/7 * 5/6) + (5/7 * 2/6) = 0.47619047619 or 47.619047619%

Isn't it interesting how two cards in our hand is just as likely as one and zero cards is exactly 10x less likely than either of those?

(.47619047619 + .47619047619 + .04761904761) = .999999999999

We have accounted for all possibilities.

With that, let's bring our table back down:

Probability of Full House: 0.025961 Return on FH: .025961 * 11 = .285571
Probability of 4OaK: 0.001681 Return on 4OaK: .001681 * 75 = 0.126050
Probability of SF: 0.000279 Return on SF: .000279 * 250 = .069627
Probability of Loss: -0.972047

TOTAL THUS FAR: -0.491156

Now, we understand that we are to be paid 1000 units for a RF using community cards, so here is that:

1000 * (.000032 * (.47619047619 + .04761904761)) = 0.01676190476

So, we combine the return of that with the total so far: 0.01676190476 - .491156 = -0.47439409524

Therefore, to have a break even jackpot, the return of the jackpot must be .47439409254 units. That is a simple algebra problem:

(.000032 * .47619047619) * x = .47439409524 (Solve for X)

x = 31,132.1125

Therefore, the value of the Progressive must be $31,132.1125 for you to have a break even proposition. Anything in excess of that is an advantage.

You might want to know how that applies to the base game. That's simple. If you know the Element-of-Risk for the base game, then you simply express that bad boy as a decimal, multiply that by the average units bet per play and then multiply that by how much a unit is worth. After you have done that, add it to the .47439409524 from the equation above and resolve. Here's a freebie:

https://wizardofodds.com/games/heads-up-hold-em/

For one paytable, the EoR is 0.64% or .0064, and the average total bet is 3.67 units, if we make a unit $5, here is what we get:

5 * 3.67 * .0064 = 0.11744

That means you can expect to lose 11.744 cents per play, Wizard has it at 11.79 cents, I think that's due to rounding, so we will go with the greater of the two. If we add .1179 to our .47439409524, then we see that the contribution of Royal w/ both cards must now be .5922940952 to cover our expected loss, so we need to replace that in the equation.

(.000032 * .47619047619) * x = .5922940952

x = $38,869.299998, essentially $38,869.30

In order to have a break even play, assuming the side bet is $1 and the minimum for the game is $5, you would need the Progressive to be at $38,869.30. Now, if the minimum is $10, then just add another 11.79 cents to the right side of the equation, if it is $15, then add .2358 to the right side of the equation and redo.

Thanks you sir

You're welcome, and remember, there is some rounding here and there. Also, I believe that the EoR of your base game paytable is different, but you can look up the EoR and change the equations I have provided accordingly.