Analysis of Triple-Play and other multi-line VP?

In an effort to take advantage of certain promotions at a casino in a nearby town, I have been trying to determine the expected cycle of certain hands, as well as ROR calculations for Triple-Play. The best $1 game in the house is Triple-Play Super Aces Bonus 8/5 with a distant second being $1 single hand Bonus 8/5. This casino issues one of the following coupons each month:

200-Coin Bonus for Aces or Faces (natural four of a kinds)
100-Coin Bonus for any natural Four of a Kind
100-Coin Bonus for a Dealt Flush
100-Coin Bonus for a Hearts Royal Match (HRM is the Ace of Hearts and at least one suited King, Queen, or Jack on the deal)
100-Coin Bonus for a Spades Royal Match (SRM is similar to above)

The last three bonuses in the list above are easy to determine, as the cycle length is independent of the game or strategy. The first two are giving me problems, as I do not believe that the cycle length of at least one quad on Triple-Play is that of the single line game divided by three. While it is close, it will be longer, due to the possibility of having more than one quad at a time (no extra bonuses!).

I completed a test to see if calculating the total outcomes of three hands in a row and their respective probabilities (average the individual outcomes and multiply the individual probabilities), and I arrived at the same return (as a checksum). The problem is that I get an expected cycle for 3 Royals being 60,006,373,247,950 (the expected cycle for a dealt Royal is only 649,740), so without checking any other combinations, I know the mix for three single hands in a row is not the same as that of three at a time.

Does anyone have a "simple" trick to determining the outcomes and probabilities for multi-line play?

It seems that this is something that could be added as an afterthought to the paytable calculators - once each hand is evaluated for its optimal play, the results for 3, 5, 10, etc. hands could be tallied along with those for single line play. Unfortunately, my programming is rusty (I used a spreadsheet to calculate FP Pick'Em!)

I was afraid I'd hear crickets on this one! lol Fine, I'm taking my ball and going home, 'cause I don't want to play with youse kids anymore!



...See ya' tomorrow!

The first step would be to calculate the adjusted strategy. If you use the video poker calculator, you can use Super Double Bonus to get the extra payline you need. Plugging in the 8/5 Super Aces payoffs with the bonuses gives the following adjusted paytable:

Hand	Coins
Royal Flush	4000
Straight Flush	300
Four Aces	2200
Four Js, Qs, Ks	450
Four 2s, 3s, 4s	500
Four 5s thru 10s	350
Full House	40
Flush	25
Straight	20
Three of a Kind	15
Two Pair	5
Jacks or Better	5

... that alone returns 106.3712%. You can also use this paytable in the strategy generator. I already did and the strategy is wicked easy and has very few exceptions.

Before answering your question, I want to figure out the value of the deal bonuses. I'll do the Royal Matches first, since a dealt Flush might include a Royal Match, which would need to be subtracted from the Flush total. A Royal Match per your description could be made in the following ways:

Faces	Suited Cards	Unsuited Cards	Flush?	Combinations
1	3	0	Yes	504
1	2	1	No	8424
1	1	2	No	40014
1	0	3	No	54834
2	2	0	Yes	216
2	1	1	No	2106
2	0	2	No	4446
3	1	0	Yes	18
3	0	1	No	78
Total				110640

So 110,640 hands have a Royal Match. Of these, 504+216+18=738 of them have a Flush (or Royal Flush).

Now, the number of flushes on the deal (including dealt Straight Flushes and Royal Flushes) is combin(4,1) * combin(13,5) = 5148. Of these, 738 have already been accounted for, so 5148-738 = 4410 remain unaccounted for.

So now we know that 110640+4410 = 115,050 starting hands have some sort of Flush and/or Royal Match and are worth 100 coins each. I assume you only get the 100-coin bonus once, even though you are playing three hands simultaneously. If that is correct, the added return is 100 * 115050 / (15 * 2598960) = 29.5118%. If the bonus pays 100 coins on each hand, the added value is 100 * 115050 / (5 * 2598960) = 88.5354%.

So you are looking at a total player advantage of either 35.88% or 94.9%, depending on how the deal bonuses work. Either way, you are looking at a massive player advantage!

As to your question about figuring out the probability of getting a quad on a draw hand in triple-play, that's more involved. In order to determine that, you need to look at the probability of being dealt a hand which, when played optimally, can result in a quad on the draw. So for example, being dealt 4 to a royal is a hand which cannot produce a quad on the draw if you are playing optimally. Holding three of a kind is more likely to produce a quad than holding a pair, etc.

So you would have to look at all the hands which can yield a quad when played properly, and for each such hand, figure out the probability of ending up with 1, 2, or 3 quads on the draw, and tally the results for the game as a whole. I would say that such a task is of medium-low difficulty, but I don't have the time to figure it out right now (sorry). If you were playing single-hand, you'd be looking at an average of about 416.8 hands to get a single quad. In triple-play, it depends more heavily upon the decision hand. I don't think there is a quick formula to compute it; you'd need to look at every deal hand in more detail than is normally done.

However, who cares? The dealt Flushes and Royal Matches are where the bulk of the EV is coming from, occurring once every 22.6 deals on average. Get to that casino and crush this before they kill it! You could even play 6/5 Jacks and still make a very hefty profit.

(Hopefully I did the math on this correctly. I'm sure someone will let me know if I did not.)

Quote: JB

a hand which cannot produce a quad on the draw if you are playing optimally. Holding three of a kind is more likely to produce a quad than holding a pair, etc.

So you would have to look at all the hands which can yield a quad when played properly, and for each such hand, figure out the probability of ending up with 1, 2, or 3 quads on the draw, and tally the results for the game as a whole. I would say that such a task is of medium-low difficulty, but I don't have the time to figure it out right now (sorry). If you were playing single-hand, you'd be looking at an average of about 416.8 hands to get a single quad. In triple-play, it depends more heavily upon the decision hand. I don't think there is a quick formula to compute it; you'd need to look at every deal hand in more detail than is normally done.

However, who cares? The dealt Flushes and Royal Matches are where the bulk of the EV is coming from, occurring once every 22.6 deals on average. Get to that casino and crush this before they kill it! You could even play 6/5 Jacks and still make a very hefty profit.

(Hopefully I did the math on this correctly. I'm sure someone will let me know if I did not.)

I agree. The most valuable coupons have very little of a cycle. And this is very naive, but a rough assumption would be to factor out extra quads from dealt quads and dealt trips. And I think your table is right, but I would have never came up with doing it that way. :(

Probability of Dealt Quads: 1 in 4165

Probability of Dealt Trips (and Aces Full):
{12*C(4,3)*[C(48,2) - 12*6] + C(4,3)*C(48,2)}/2,598,960 = 0.0212392649

Chances of making quads from Trips:
46/C(47,2) = 46/1081 = 0.04255319

From Binomial Distribution:
Zero quads from trips: 0.8776956936
One quad from trips: 0.1170260925
Two quads from trips: 0.0052011597
Three quads from trips: 7.7054217E-005

Initial bottom line probability to get quads: 1 in 416.8 rounds
Duplicates from Dealt Quads: 1/4165 * 2 = 1 in 2082.5
Duplicates from Dealt Trips after the draw: 0.0212392649*(0.0052011597 + 2*7.7054217E-005) = 0.00011374 = 1 in 8791.8

Approximate average number of hands needed to get a quad without counting duplicates:
1/416.8 - 1/2082.5 - 1/8791.8 =~ 1 in 554.

I didn't do the exact math on counting dealt pairs, but it looks like it it's fairly negligible: If holding a pair is the correct play 50% of the time (overestimate), the final answer changes to ~1 in 557.5 hands played, or ~1 in 185.83 rounds played.

Thank you for the response, JB! I appreciate and value the methods offered. I, too, have been using the calculators for Super Double for that very reason (another casino that I frequent gives coupons for quads, but only 5's thru T's), and as long as the game is set up like Jacks or Bonus it works like a charm! Obviously, kicker games require a bit of approximation...

As for the coupons, I wish it was the way you calculated! Unfortunately, the deal isn't as good as that - sorry for my hasty description! Each month the property chooses only one of the coupons from the list below:

200-Coin Bonus for Aces or Faces (natural four of a kinds)
-OR- 100-Coin Bonus for any natural Four of a Kind
-OR- 100-Coin Bonus for a Dealt Flush
-OR- 100-Coin Bonus for a Hearts Royal Match (HRM is the Ace of Hearts and at least one suited King, Queen, or Jack on the deal)
-OR- 100-Coin Bonus for a Spades Royal Match (SRM is similar to above)

Currently, the coupon offered is 200-coins for Aces or Faces; last month was 100-coins for a natural quad; and the month prior was 100-coins for a Royal Match in Spades.

For either Royal Match (spades or hearts), I calculate an expected cycle of 36.3 hands for any five-card-draw game using a 52-card deck. This is how often you should be dealt two or more to a Royal including the Ace in a given suit . If memory serves, a particular Ace (or any other single card) should appear on the deal once every 10 hands or so. This coupon has an expected profit of $99.67 when playing $1 Triple-Play SAB 8/5 or $98.48 when playing $1 single hand Bonus 8/5. Even playing a 90% game would give you an expected profit over $81, as long as you play dollars! ...Extremely Crushable!

For a Dealt Flush, I calculate (or look up) an expected cycle of about 505 hands under the same conditions as above. This includes dealt straight flushes and royals, as the coupon indicates five natural cards of the same suit on the deal. This coupon has an expected profit of $95.39 when playing $1 Triple-Play SAB 8/5 or $78.95 when playing $1 single hand Bonus 8/5. I usually wimp out and play single hand due to the volatility of SAB, especially at $15 a pop! ...Somewhat Crushable

For any natural quad, I see an expected cycle of about 423 hands on Bonus 8/5 without adjusting strategy for an expected profit of $82.35. For Aces or Faces on Bonus 8/5, the cycle is 1,278 hands resulting in an expected profit of $146.72.

Single hand SAB 8/5 has an expected cycle for any quad of about 417 hands and for Aces and Faces about 1,225 hands. Dividing either of these by three for Triple-Play is close, but not accurate, thus the discussion at hand.

As the profit will likely be about $15 higher for quads and about $50 higher for Aces and Faces, I would like to know more about ROR implications, as well as true cycle lengths, since these opportunities don't come around very often.

Quote: tringlomane

I agree. The most valuable coupons have very little of a cycle. And this is very naive, but a rough assumption would be to factor out extra quads from dealt quads and dealt trips.

Thank you, as well, tringlomane, for your response. I see JB's point about the exercise being medium to low difficulty (or you both just made it look easy!)

I had considered this approach, but got caught up with trying to make the same calculations for pairs and other two-card holds, single high-card holds, and redraws. Guess I hadn't gotten far enough to rule them out due to extreme unlikelihood. (How awesome would it be to hold a single Ace and receive not one, but two or three quads?!?) My strong math and statistics background tends to drive me to attempt exact answers, even when there isn't much of an affect to practical use. Sometimes I need to pull back by using a Pareto analysis - 20% of the work gets you 80% of the way to an accurate answer. Talk about diminishing returns, and I'm not even at the casino yet! lol

Quote: JB

So you are looking at a total player advantage of either 35.88% or 94.9%, depending on how the deal bonuses work. Either way, you are looking at a massive player advantage!

Unfortunately, 100 coins is the total bonus, regardless of the number of successes (such as two or more guads). Do you think they'd accept a double coupon from the grocer?!?

Quote: camapl

I, too, have been using the calculators for Super Double for that very reason (another casino that I frequent gives coupons for quads, but only 5's thru T's), and as long as the game is set up like Jacks or Bonus it works like a charm! Obviously, kicker games require a bit of approximation...

Hey JB, what is the probability that you implement the "Custom Video Poker Game" calculator of VPGenius fame on the WoO site?

It occurred to me this morning that I could calculate ROR using a $15 single hand of SAB 8/5 as an upper limit of the risk I would partake while playing three $5 hands at a time. Say for my current bankroll, I calculate a 1 in 5 chance of ruin, then I would know that my chances of success are even better. After all, these are just confidence limits, which are subject to the same stochastic process as the game itself. Thanks again for steering my thinking straight!