How do you count close calls to Royal Flush?

[$licktwo]

Hello players. I play a ton but always use math based advantages and strategies. I only play single line dw and job. I like to count how many hands and how many 4 to a Royal I get between actual Royals. It helps me carry on through the grind between Royals

A close call is when I have 4 to a royal on screen and the 5th card is a random card.
Dealt 4 to a royal is a close call. Drawing and missing is a second close call.
Holding 9jk suited and getting aq suited is not a close call.

Using this method I had a 588 close calls and 220,000 hands between my last two royals. Shouldn't the average be 47 close calls because that is how many cards the 5th could be?

My question is what is the expected number of close calls in say 40,000 hands?

[Player422738]

Dealt 4-to-a-royal happens every 2765 (47*5*4 / 2598960) hands on 52 deck game. To complete the Dealt 4 to a royal u have 1 in 47 chance. So you can complete the Dealt 4 to a royal every 130K hands.

The Dealt Royal happens every 235 Dealt 4 to a royal.

Of course there are other ways to complete the royal and that makes the the Royal happen every 40K hands on average.

Last but not least, Royal is not due. Your statistics doesn’t change the odds. So there is no need to count anything. The game is fair in terms of randomness.

[Jstark]

"A close call is when I have 4 to a royal on screen and the 5th card is a random card."

No matter what card you draw, barring a class II machine, it's always random.

[Eduardo]

So it sounds like you also want to know the frequency of a 4 to the royal on the final hand (after the draw).

I don't think I've ever seen that number. I mean, other than the specific record keeping you are talking about, it is fairly meaningless. But I'm interested if anyone can come up with it.

I know it's not 47 times the frequency of a royal, despite the reasoning that the 5th card could 1 in 48 cards when 4 of them are "royal" cards. That is only the odds when holding 4 to the royal and ending up with 4 to the royal.

But you are going to have a LOT more "close calls" than that. Holding 3 to the royal, you now have 2 cards to draw and only one of them has to be a royal card for it to be a "close call," with 2 in the deck.

So that's 2/47 odds of the first draw card being a royal and 2/47 for the second (minus the odds of both of them being a royal. Your odds of ending up with 4 to the royal holding 3 are quite good. Your odds of HITTING a royal holding 3 are quite bad.

Drawing 3 cards is more complicated, because now you need to get 2 out of the 3, but you have 3 cards to draw into and 3 matching cards in the deck.

Much more math than I care to get into at the moment. But the bottom line is, you should expect a LOT of close calls. Much more than 47. Your 588 may not be far off from expectations.

[billryan]

I forget the exact math so lets say 4 to a Royal is worth $17.50. You could celebrate every time you hit it, adding an imaginary $17.50 to your mental BR. That builds in the times you will go on to hit a royal so you can have 47 small celebrations or one large one.

[onemoretry]

My take is that there are 940 combinations of cards that create a four to the royal. For each suit, there are 5 ways to arrange four of the royal cards, and for each arrangement, there are 47 non royal cards. So, it's 5x47x4=940. (God, I hope I didn't screw this up!).

[$licktwo]

Thanks for the responses. I thought that one would average 47 close calls between actual Royals. I knew I had to be mistaken when I got nearly 600 close calls! Thanks for clarifying.

I said a close call involves the 5th card bring a random card. I should have said a non-held card. I believe all the cards are random. If they weren't random the math would be worthless and I would not play.

[New2vp]

Of course, if you want an exact number, each game (Jacks or Better or Deuces Wild) is different as to the number of "close calls" as you have defined it. And there would also be differences between the various pay schedules. Below is a sample calculation for 9/6 Jacks or Better if I understand your definition of a "close call."

If my calculations and estimates are close, I get an expectation of about 527.28 close calls in 220,000 hands. I further estimate that the standard deviation is only about 26.10. 588 is a bit more than 2.3 standard deviations from my expected value, but I of course don't know how exact your 220,000 figure was and I do not have a guess how the actual distribution of the games you played might affect the number. If you were playing error-free 9/6 Jacks or Better for 220,000 hands, 588 close calls would be on the high side, but not totally out of the question.

I think it is more appropriate to compare 588 close calls with the 220,000 total hands than to somehow look at a ratio of 588 to only 2 royals because you clearly had a bit of a royal drought in the sample you discussed.

I certainly agree with Eduardo's assessment and rationale why there are considerably more than 47 close calls per royals in the long run. Below, I'll show my estimate (about 97) and how I reached it.

Onemoretry's calculation is spot-on. 940 dealt 4-to-the-royals. And on 936 of them, there is a 46/47 chance that you will have a 2nd close call on the same dealt hand according to the definition provided. 1/47 of the time, you will get a royal, so there would only be a single close call, that being the one on the deal. The other 4 dealt hands are the KQJT9 straight flushes of the four suits. With perfect strategy, you will not get a 2nd close call in my calcs below because that hand would not be drawn to.

In a cycle of the 2,598,960 possible hands, I would expect 6228.95 close calls. Video poker software suggests that you would get 64.3457 royal flushes, so the ratio of the expected values would be 96.80 close calls for each royal. Of course, when you have a royal drought, the ratio would be expected to be larger. And with a glut of royals (more than anticipated given the number of hands), the ratio could be expected to be considerably smaller.

To answer one of the questions directly, in 40,000 hands, I would expect 95.87 close calls.

Below, I mostly used Video Poker for Winners to get the counts. I extended the method that Eduardo hinted at to all possible holds that could result in a royal flush. I will indicate where I made estimates, a couple of which could contain some inaccuracies, though hopefully small ones.

Close Calls in 2,598,960 dealt hands:

1. Dealt 4 to the Royal
940 x 1 = 940
2. Held RF 4
936 x 46/47 = 916.09
3. Held RF 3
27,492 x 90/1081 = 2288.88
4. Held RF 2
204,660 x 132/16,215 = 1666.06
5. Held J,Q,K,A
403,968 x 172/178,365 = 389.55 (Close Calls of suit held)*
6. Adjustments to #5
12,996 x (-172)/178,365 = -12.53**
7. Draw 5 (Discard a ten)
38,665 x 630/1,533,939 = 15.88***
8. Draw 5 (No ten dealt)
45,695 x 840/1,533,939 = 25.02

This totals to 6228.95

*For example, when holding the J of hearts, one can draw AKQJ of spades. This would be an example of a final hand 4-to-a-royal that was not a close call, since you cannot get a spade royal when holding a heart.

**The point here is to adjust downward when it is impossible to draw a royal. I did some counting to find out how often one holds a single high card (a King or an Ace) and discards the Ten of the same suit. VPFW seems to show 1440 instances of holding the King and tossing a suited ten. It requires more calcs for determining how often one holds an ace and tosses a suited ten. I counted 11,556 instances. This could be off, but the effect as shown above for both KT and AT is small.

***VPFW shows 84,360 hands in which 5 new cards are drawn. There are 126 combinations of 5 ranks from the set {2, 3, 4, 5, 6, 7, 8, 9, ten}. Six of these are straights (like 5432 or 9876). Of the remaining 120 combinations, 55 of these contain a ten while the other 65 do not. When there is no ten discarded, you can royals in all 4 suits. When a ten is discarded, you can get a royal only in the other 3, so you can't get close calls in the ten's suit. So I allocated the 84,360 hands with 55/120ths of them assumed to have a ten and the other 65/120ths not having a ten and having a better chance at getting a close call.

Sorry about the length of the post, but I guess I jumped at the bait of Eduardo's challenge of coming up with a number. Since I don't have a program, I would not want to try this for other games or pay schedules. If anyone has further interest, maybe I've shown a way to accomplish it.

[$licktwo]

Wow. I am dumbfounded. I thought I was the only person interested in this matter.

Specifically the two games I play are 9-6 job with RF, SF, and 4oak progressives and 10-6 dw with RF progressive. $.50 or $1 denom. Single line.

Winstar and Choctaw in OK

[advantage playe]

hey new2vp, are you sure you are new to video poker ?? impressive most impressive !!!!!!!!!!!!