Return On Dealt Royal Flush Progressives

[dw44]

Here's another way of looking at it.

One dealt royal cycle is 649,741 hands.

Wagering $2.50 per hand we would coin-in $1,624,352.5 for one such cycle and expect on average a return of .9954 as much, or $1,616,880.4785, for an expected loss of $7,472.0215 per cycle.

So we need an extra $7,472 per dealt royal on top of the usual $2,000 per royal. That comes to a total of $9,472 per royal, which means $47,360 to cover five simultaneous royal.

I don't know, I may be misunderstanding something. But it does seem like the rarity of dealt royals means we really need a big surplus.

Consider that using 9/6 optimal strategy we get a royal (dealt or drawn) 1 in 40,391 hands.

So a surplus that applies to only dealt royals is worth about 6.2% what a surplus that applies to all royals is worth (ignoring strategy deviations which could make the disparity larger).

[Vman96]

"Royals on All Lines" Jackpots add a lot less to the return than most people think. It's a great feature for casinos to offer because players are more willing to play them even with inferior paytables.

The dealt royal probability is 1 in 649,740 by the way, or 649,739 to 1. But on these units, technically you should also consider the minute chance of you getting dealt 4 to a royal and converting all five lines (1 in 634,138,944,355). This actually improves your chances of getting the jackpot to 1 in 649,739.334. Of course you could convert five of five 3 to a Royals as well and so on, but it will barely change the overall odds. Heck, 4 to Royal barely did.

The probability of any Royal is about 1 in 40,390.55, but the probability of 5 of 5 Royal only is about 1 in 649,739.334 just over 16 times more rare. So doubling the dealt royal value should roughly only add about 0.02/16 = 0.00125 to the return. But let's also look at the exact math to break even.

The base return of the game ($10k jackpot for 5 royals) is 0.995439.

The extra return from getting five of five royals is

(X-800)/649,739.334 where is is the number of $12.50 betting units and the $10k jackpot is your standard 800 for 1 Royal on each line.

So to break even, you need (X-800)/649,739.334 to be equal to 1 - 0.995439

(X-800)/649,739.334 = 1 - 0.995439

X - 800 = 2963.4611

X = 3763.4611 units

3763.4611 units x $12.50/unit = $47,043.26

My number is slightly different than dw44 because he divided by 649,741 instead of the number I calculated above and well as me using 99.5439% return instead of 99.54%.

[Vman96]

Here's another way of looking at it.

One dealt royal cycle is 649,741 hands.

Wagering $2.50 per hand we would coin-in $1,624,352.5 for one such cycle and expect on average a return of .9954 as much, or $1,616,880.4785, for an expected loss of $7,472.0215 per cycle.

So we need an extra $7,472 per dealt royal on top of the usual $2,000 per royal. That comes to a total of $9,472 per royal, which means $47,360 to cover five simultaneous royal.

I don't know, I may be misunderstanding something. But it does seem like the rarity of dealt royals means we really need a big surplus.

Consider that using 9/6 optimal strategy we get a royal (dealt or drawn) 1 in 40,391 hands.

So a surplus that applies to only dealt royals is worth about 6.2% what a surplus that applies to all royals is worth (ignoring strategy deviations which could make the disparity larger).

You got the mechanics of calculating it right, the probability of a dealt Royal is 1 in 649,740 though. There won't be strategy deviations with this game. The vast majority of the jackpot bonus comes from a hand the machine will auto-hold.   

[dw44]

Thanks for the assistance. In my last aside I was thinking of possible strategy deviations for the hypothetical all-royals jackpot which would make it even more valuable compared to the dealt royals jackpot.

[Vman96]

Thanks for the assistance. In my last aside I was thinking of possible strategy deviations for the hypothetical all-royals jackpot which would make it even more valuable compared to the dealt royals jackpot.


Yeah, but you need to have 3 to a royal to even possibly want to deviate strategy, since the standard play is to hold 4 to a royal. The probability of converting five of five 3 to a royal draws is 1 in 1081^5, which is 1 in 1,476,143,130,389,401...so that's probably never happened in the history of Video Poker.

[Kshman]

Sounds good. Thank you Vman and dw44. I really appreciate the help and effort.

[BobDancer]

To simplify, I calculate it in terms of "double royals" -- i.e. 800 more coins every 650,000 dealt hands.800/650,000 = 0.12%So a double dealt royal ($20,000 in your example) pays 99.54% + 0.12% = 99.66%A triple dealt royal ($30,000 in your example) pays 99.54% + 0.24% = 99.78%Any dealt royal amount between these figures may be easily approximated.Ignore the chance for getting 5 royals by drawing one card to five hands. Yes it's possible --- but it's a rounding error. I was already using 650,000 rather than 649,960 and the 0.12% is rounded as well. This is a case where there's no value to more precision. You're talking a few pennies or dollars in maybe 800 hours of play.Ignore strategy changes. The only time you wouldn't hold 4-to-the-royal is when you have a king-high straight flush. This progressive is nowhere near high enough.The technique of "double the value of the dealt royal and it pays an additional 0.12%" works on any game, with any number of lines, so long as it has a 52-card deck.

[Kshman]

Thanks Bob. I appreciate the help.