So far, 2021 has been pretty good to me as far as VP is concerned. I’ve been very fortunate to have drawn four Royal Flushes (RF) and seven Four Aces with a kicker (4AWK) in the first three months of the year. Of the seven 4AWK, six have been on TTB and one on Triple Double Bonus (TDB). While I enjoy playing all the different VP variations, I’m particularly fond of TTB, TDB, Jacks or Better (JoB), and Deuces Double Double Bonus (DDDB). All of my of my RFs came while playing JoB, DDDB, Deuces Wild, and Deuces Bonus. I spend most of my time playing TTB compared to the other VP games, yet I have not drawn a single RF on TTB this year. I understand that the odds of drawing an RF are about one in 41,000 hands, depending on the VP game. I assume that the odds are about the same for TTB, so I can understand not pulling an RF on TTB in just three months. What I don’t understand is the high number of seven 4AWKs compared to zero RFs on TTB and TDB. This leads to me to ask: What are the odds of drawing 4AWK on TTB? Are they the same for TDB? I assume that the odds are the same and that a 4AWK is mathematically more likely than a RF on both TTB and TDB. Still, I find that pulling 7 4AWKs compared to 0 RFs on TTB and TDB interesting. Should I chalk it up to coincidence and this is not outside the standard deviation?

I also have a question about the pay tables on TTB and TDB: if the odds of pulling a 4AWK are better compared to a RF, why is the payout the same for both? 4,000 credits for an RF and 4,000 credits for 4AWK (as well as 4,000 credits for 4 2s, 3s, 4s with an Ace on TTB) are great, but I don’t understand why they pay the same. Does it have something to do with the lower pay outs for lesser hands?

Quote:PaboHello, all. I haven’t been active for a while, but I have been lurking. I’ve learned a lot from the experts here on Wizard of Vegas when it comes to gambling-related discussions. I’ve also learned a lot from Wizard of Odds. Thank you, Mr. Shackleford, for both sites, which have proven very informative and helpful for me over the years. I have some questions for the forum regarding video poker (VP), particularly Triple Triple Bonus (TTB). Frankly, I’m mathematically challenged. I’m strong in other subjects, but math isn’t one of them. I’ve thought about this issue for a while, but have been unable to figure out the mathematical answer for myself. I’m hoping that someone here can help me.

So far, 2021 has been pretty good to me as far as VP is concerned. I’ve been very fortunate to have drawn four Royal Flushes (RF) and seven Four Aces with a kicker (4AWK) in the first three months of the year. Of the seven 4AWK, six have been on TTB and one on Triple Double Bonus (TDB). While I enjoy playing all the different VP variations, I’m particularly fond of TTB, TDB, Jacks or Better (JoB), and Deuces Double Double Bonus (DDDB). All of my of my RFs came while playing JoB, DDDB, Deuces Wild, and Deuces Bonus. I spend most of my time playing TTB compared to the other VP games, yet I have not drawn a single RF on TTB this year. I understand that the odds of drawing an RF are about one in 41,000 hands, depending on the VP game. I assume that the odds are about the same for TTB, so I can understand not pulling an RF on TTB in just three months. What I don’t understand is the high number of seven 4AWKs compared to zero RFs on TTB and TDB. This leads to me to ask: What are the odds of drawing 4AWK on TTB? Are they the same for TDB? I assume that the odds are the same and that a 4AWK is mathematically more likely than a RF on both TTB and TDB. Still, I find that pulling 7 4AWKs compared to 0 RFs on TTB and TDB interesting. Should I chalk it up to coincidence and this is not outside the standard deviation?

I also have a question about the pay tables on TTB and TDB: if the odds of pulling a 4AWK are better compared to a RF, why is the payout the same for both? 4,000 credits for an RF and 4,000 credits for 4AWK (as well as 4,000 credits for 4 2s, 3s, 4s with an Ace on TTB) are great, but I don’t understand why they pay the same. Does it have something to do with the lower pay outs for lesser hands?

1.) Overview:

You have that right when you say, "Depending on the VP game," but it also depends on the VP paytable for the same game and assumes that you're following optimal strategy.

For reference, here's the Triple Triple bonus page:

https://wizardofodds.com/games/video-poker/tables/triple-triple-bonus/

If you look towards the top, when the flush goes from 6 to 5, not only does the return percentage drop significantly, but flushes/royals also become considerably less likely given optimal strategy.

On 9/5 Triple Triple bonus, the final hand probability for AWK seems to be 0.00007100133 which means it is expected to happen one in every 14084.24 hands.

Again, that assumes optimal strategy for that game, so any strategy deviations will result in probabilities that, effectively, are slightly different.

You'll also note that this makes AWK almost three times more likely than the Royal, but if you switch the paytable to 9/6 TTB, then the Royal becomes a bit more likely and flushes become considerably more likely.

When we look at Double Double Bonus, and again, this is going to vary based on paytable and corresponding strategy, but for 9/6/4 TDB, AWAK is 0.0000703949 or about 1 in 14205.57, which is pretty close to the other game. I don't know for sure, but would intuit that the difference comes from perhaps holding an Ace with one other high card on the 9/6/4 (flushes pay more than above as do straights) where you wouldn't do that on TTB---or, are at least not supposed to.

Actually, I think it might be high straights making the difference as well as some straight flush draws on the low end that take preference over holding a single Ace on TDB. I don't know for sure as TTB doesn't seem to be available on the strategy maker. I'm more confident about the high straights (single ace preferred sometimes) than I am about the low end straight flushes.

Anyway, the probabilities for AWAK are close enough that, given optimal strategy, you probably wouldn't even notice the difference from one game to the other in the practice of individual play.

2.) Payouts:

It has precisely to do with the lower payouts for lesser hands, for comparison, would you play Jacks or Better if 2P broke even, 3OaK paid 2-For-1 and a straight paid 3-For-1 instead of 4-For-1? You'd get massacred. In fact, let's look at 9/6 JoB, but we will change straights on down to reflect TTB.

And, with that, Jacks or Better becomes a 78.21% (rounded) return-to-player game. There are people who would probably play it.

3.) Other Stuff:

As far as your results are concerned, I could give you more insight if I knew how many hands you played this year, but I don't. Let's see...we know that the AWAK is about three times more likely than a RF and that you have seven AWAK to zero RF, so here's a comparison:

Imagine a Roulette wheel where we ignore zeroes, so there are just the 36 non-zero numbers. AWAK represents numbers 1-27 whilst RF represents numbers 28-36. In this case, we have seen numbers 1-27 hit seven times to zero times for numbers 28-36.

(27/36)^7 = 0.13348388671

In other words...and this is a very rough way of comparing (it's not exactly right) the probability that you might see seven AWAK before a Royal and assuming Optimal Strategy is probably somewhere between 10-15% likely, so that's not weird at all.

Actually, we can do this for TDB (or both) really quick using absolute probabilities. Check it out!:

https://wizardofodds.com/games/video-poker/strategy/a-1-b-120-c-1-d-0-d-1-d-1-d-2-d-4-d-6-d-9-d-50-d-80-d-160-d-400-d-800-d-50-d-800/

Okay, so that's 9/6/4 TDB. The first thing we need to get is a more precise RF probability and one for the AWAK:

RF: 501,972,216 /19,933,230,517,200 = 0.00002518268

AWAK: 1,403,197,800/19,933,230,517,200 = 0.0000703949

Okay, so how this works is that we are going to ignore everything else and pretend that these are the only two possible hands, or, at least, the only two we care about:

0.0000703949+0.00002518268 = 0.00009557758

AWAK: 0.0000703949/0.00009557758 = 0.73652105441

RF: 0.00002518268/0.00009557758 = 0.26347894558

Right, so we want the probability of having seven AWAK before we have a RF and we are looking at the two things in isolation:

(0.73652105441)^7 = 0.117569904

About 11.757% or fewer than 1 in 9 players will have seven AWAK before they hit a RF given any starting point.

Definitely a fun problem to figure out, but it's so much within the realm of possibility that I wouldn't even call it coincidence.

(This post has been edited due to a small copy/paste error in #3 that did not change the final result of #3.)

That's basically the long and short of it, not a terribly unusual occurrence. And, of course, these quads having already happened...it's just as likely (about 1 in 9) that you'll get another seven quads before any Royal Flush, which would then be fourteen before any Royal.

That's a shame about the VP machines, it's not much different around me. You have to check all kinds of machines to even find anything over 98%---and even those are very few.

Sometimes you have to think outside the box.