Percentage of 3 card Royals played
June 13th, 2015 at 7:55:49 AM
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I remember the wizard answering a question, once upon a time, that you get dealt a 3 card royal about once every sixty hands. I am assuming that he took the four card royals out of that calculation, but I don't know. I have 2 questions:
1. In a normal 4000 royal 9/6 JOB game you would hold high pair, 2 pair, 3 of a kind, straights and flushes over a 3 card royal. What percentage of the time do you end up actually playing the 3 card royal with these hands removed (assuming you are playing as you statistically should)?
2. When the RF progresses to around 5700 and above you start playing the 3 card royal over the high pairs (you do different groups as it goes up). What percentage of the time are you then playing the 3 card royal?
Thanks for any consideration.
1. In a normal 4000 royal 9/6 JOB game you would hold high pair, 2 pair, 3 of a kind, straights and flushes over a 3 card royal. What percentage of the time do you end up actually playing the 3 card royal with these hands removed (assuming you are playing as you statistically should)?
2. When the RF progresses to around 5700 and above you start playing the 3 card royal over the high pairs (you do different groups as it goes up). What percentage of the time are you then playing the 3 card royal?
Thanks for any consideration.
July 13th, 2015 at 7:40:18 AM
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Wish I had an answer for you - but it's a great question!
"Those who have no idea what they are doing, genuinely have no idea that they don't know what they are doing." - John Cleese
July 25th, 2015 at 7:48:49 PM
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I would title my reply as "any" consideration - meaning, it's not backed by math.
For me - it depends. I don't care how high the royal is, I'll only play a 3 card royal if the rest of the hand has no value.
Meaning, if I have
Ace of clubs
Jack of clubs
10 of Clubs
10 of diamonds
3 of spades
I think about like this: As it stands - my 5 credits are in and I've lost, unless the hand improves.
There are 3 Jacks, 3 Aces, 2 10's and 3 3's left in the deck. But, I can't keep them all. Gotta drop the 3, at least.
There are also 10 more clubs.
47 cards left.
Do I go to improve the pair of 10's, or go for a flush of any kind?
It seems more likely I'll get a second pair than another 10.
Odds of getting 2 out of the 10 clubs remaining are probably pretty good.
Odds of improving the pair of 10's are much better - I could get another 10. I could get 2 tens. Or another pair. In fact, I'm not sure of the math, but I think the odds of getting a second pair are higher than getting trips or a quad.
So - me, I hold the low pair over the 3 card royal - unless I just feel lucky. I can probably count on one hand the number of winning hands I've sacrificed in pursuit of a royal. I've never gotten a royal for that sacrifice. Plenty of flushes, but never a royal.
Just my two worthless pennies.
For me - it depends. I don't care how high the royal is, I'll only play a 3 card royal if the rest of the hand has no value.
Meaning, if I have
Ace of clubs
Jack of clubs
10 of Clubs
10 of diamonds
3 of spades
I think about like this: As it stands - my 5 credits are in and I've lost, unless the hand improves.
There are 3 Jacks, 3 Aces, 2 10's and 3 3's left in the deck. But, I can't keep them all. Gotta drop the 3, at least.
There are also 10 more clubs.
47 cards left.
Do I go to improve the pair of 10's, or go for a flush of any kind?
It seems more likely I'll get a second pair than another 10.
Odds of getting 2 out of the 10 clubs remaining are probably pretty good.
Odds of improving the pair of 10's are much better - I could get another 10. I could get 2 tens. Or another pair. In fact, I'm not sure of the math, but I think the odds of getting a second pair are higher than getting trips or a quad.
So - me, I hold the low pair over the 3 card royal - unless I just feel lucky. I can probably count on one hand the number of winning hands I've sacrificed in pursuit of a royal. I've never gotten a royal for that sacrifice. Plenty of flushes, but never a royal.
Just my two worthless pennies.
"Those who have no idea what they are doing, genuinely have no idea that they don't know what they are doing." - John Cleese
July 26th, 2015 at 2:26:29 PM
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Well as far as I know this has no significant use other than my curiosity. The Wizard initial calculation that I found would have excluded 4 card royals and royals.
I decided to give it a whack and do a calculation on it. This is assuming you are playing JOB 9/6 basic strategy with a 4000 royal. I do break out 2 pair and trips below only because I play a progressive royal game and when it gets high enough I am playing the 3 card royals over a high pair. This also happens over 2 pair etc, but the royal is unlikely to get that high.
So it was a lot of multiplying and subject to human error, but this is the breakdown I got.
1. 4 card SF with no high pair, straight nor flush (like KQJ9) = .01332
2. High pair (contains 2 pair and trips) = .28917
A. 2 pair = .02498
B. Trips = .00879
3. Flush = .02590
4. Straight = .01271
Total = .3411 (34% will not be played)
Using the Wizard's numbers of 43240 (3 card royal hands) out of 2,598,960 total hands, you will end up playing a 3 card royal about 1 hand in 91.
Playing a progressive game where you will play the 3 card royal over high pairs at some point you will end up playing a 3 card royal about 1 hand in 65.
I decided to give it a whack and do a calculation on it. This is assuming you are playing JOB 9/6 basic strategy with a 4000 royal. I do break out 2 pair and trips below only because I play a progressive royal game and when it gets high enough I am playing the 3 card royals over a high pair. This also happens over 2 pair etc, but the royal is unlikely to get that high.
So it was a lot of multiplying and subject to human error, but this is the breakdown I got.
1. 4 card SF with no high pair, straight nor flush (like KQJ9) = .01332
2. High pair (contains 2 pair and trips) = .28917
A. 2 pair = .02498
B. Trips = .00879
3. Flush = .02590
4. Straight = .01271
Total = .3411 (34% will not be played)
Using the Wizard's numbers of 43240 (3 card royal hands) out of 2,598,960 total hands, you will end up playing a 3 card royal about 1 hand in 91.
Playing a progressive game where you will play the 3 card royal over high pairs at some point you will end up playing a 3 card royal about 1 hand in 65.
July 27th, 2015 at 12:56:54 PM
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Use the Wizard's VP Strategy Calculator and plug in the values for the royal. Not only will it tell you the HE of the game (over 100% if the royal is high enough) it will update the strategy to play differently as well.
Playing it correctly means you've already won.
July 28th, 2015 at 6:03:01 PM
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Well I was driving to my local casino yesterday (75 miles away) and I got to thinking again about this basically useless information. I didn't like my computations for High pair and straights so I thought of another way to attack the problem. There may be simpler methods, but here is what I did with my calculations and corrections. If anyone sees errors please let me know. Again the Wizards calculation takes out the other 2 cards that would make a 4 card royal or royal so those hands are not in the universe of 43240 hands. So there is always only 47 cards remaining to pick from.
1. 4 card straight flush with no HP, no straight nor flush. There are five 3 card royals that fit this category (KQJ9, KQT9, KJT9, QJT9 and QJT8).
What I did with each one of them is the following. KQJ9 - There are 9 face cards, 7 of the same suit and 3 Tens. 19 total cards you can not have of the remaining 46 cards. This leaves 27 cards to make the 5 card hand. I did the same for all 5 types and it comes to 144 hands. There are 4 suits so 576 total hands out of 43240 that meet the criteria. = .01332.
2. High pair which contains 2 pair and trips (except for trip Tens). I did the following on this.
There are four 3 card royals with 3 high cards (AKQ, AKJ, AQJ and KQJ). There are 9 cards that make a high pair and each of them can be with the other 46 remaining cards. So there are 414 hands for each one. Times 4 for each type gives 1656 hands per suit and with 4 suits a total of 6624 hands.
There are six 3 card royals with 2 high cards. There are 6 cards that make a high pair and each of them can be with the other 46 remaining cards. So there are 276 hands for each of the 6 royal hands giving a total of 1656 per suit and with 4 suits a total of 6624 hands.
This give a total of 13248 out of the 43240 = .30638
3. Trip tens because they are not included in the high pair. So 3/47 times 2/46 = .00278
4. Flush seems straight forward 8/47 times 7/46 = .02590
5. Straights- Here is what I did.
There are 6 Ace high 3 card royals. Each of the 2 holes have 3 cards that can be there so there are 9 straights for each royal possibility. With the 6 types you have 54 straights and 4 suits giving a total of 216 hands.
There are 3 King high royals You must have the missing card (Q, J or T) of which there are 3 to give you KQJT. Then there are 7 cards (3Aces and 4 nines) with which to complete the straight. So 21 straights per type and 3 types give 63 straights and with 4 suits there are 252 total straights.
There is one Q high royal. You can have a straight that is Ace high (3 Aces and 3 kings) 9 times. You can have a straight that is K high (3 kings and 4 nines) 12 times. You can have a straight that is Q high (4 nines and 4 eights) 16 times. So 9 + 12 + 16 = 37 possible straights per suit with 4 suits there are 148 possible straights.
216 + 252 + 148 = 616 total straights or .01425
6. 2 pair seem straight forward. 9/47 times 6/46 = .02498
7. Trips seem straight forward. 9/47 times 2/46 = .00879
So adding up categories 1 (.01332), 2 (.30638), 3 (.00278), 4 (.02590) and 5 (.01425) gives .36263 or 36.26% of the 43240 hands that you do not play. This means you play a 3 card royal about every 94 hands.
If you play a progressive royal and it is high enough to play over the High pair then if you add up categories 1, 4, 5, 6 and 7 you get .08678 or 8.68% of the 43240 total hands. This means you play a 3 card royal about every 66 hands.
1. 4 card straight flush with no HP, no straight nor flush. There are five 3 card royals that fit this category (KQJ9, KQT9, KJT9, QJT9 and QJT8).
What I did with each one of them is the following. KQJ9 - There are 9 face cards, 7 of the same suit and 3 Tens. 19 total cards you can not have of the remaining 46 cards. This leaves 27 cards to make the 5 card hand. I did the same for all 5 types and it comes to 144 hands. There are 4 suits so 576 total hands out of 43240 that meet the criteria. = .01332.
2. High pair which contains 2 pair and trips (except for trip Tens). I did the following on this.
There are four 3 card royals with 3 high cards (AKQ, AKJ, AQJ and KQJ). There are 9 cards that make a high pair and each of them can be with the other 46 remaining cards. So there are 414 hands for each one. Times 4 for each type gives 1656 hands per suit and with 4 suits a total of 6624 hands.
There are six 3 card royals with 2 high cards. There are 6 cards that make a high pair and each of them can be with the other 46 remaining cards. So there are 276 hands for each of the 6 royal hands giving a total of 1656 per suit and with 4 suits a total of 6624 hands.
This give a total of 13248 out of the 43240 = .30638
3. Trip tens because they are not included in the high pair. So 3/47 times 2/46 = .00278
4. Flush seems straight forward 8/47 times 7/46 = .02590
5. Straights- Here is what I did.
There are 6 Ace high 3 card royals. Each of the 2 holes have 3 cards that can be there so there are 9 straights for each royal possibility. With the 6 types you have 54 straights and 4 suits giving a total of 216 hands.
There are 3 King high royals You must have the missing card (Q, J or T) of which there are 3 to give you KQJT. Then there are 7 cards (3Aces and 4 nines) with which to complete the straight. So 21 straights per type and 3 types give 63 straights and with 4 suits there are 252 total straights.
There is one Q high royal. You can have a straight that is Ace high (3 Aces and 3 kings) 9 times. You can have a straight that is K high (3 kings and 4 nines) 12 times. You can have a straight that is Q high (4 nines and 4 eights) 16 times. So 9 + 12 + 16 = 37 possible straights per suit with 4 suits there are 148 possible straights.
216 + 252 + 148 = 616 total straights or .01425
6. 2 pair seem straight forward. 9/47 times 6/46 = .02498
7. Trips seem straight forward. 9/47 times 2/46 = .00879
So adding up categories 1 (.01332), 2 (.30638), 3 (.00278), 4 (.02590) and 5 (.01425) gives .36263 or 36.26% of the 43240 hands that you do not play. This means you play a 3 card royal about every 94 hands.
If you play a progressive royal and it is high enough to play over the High pair then if you add up categories 1, 4, 5, 6 and 7 you get .08678 or 8.68% of the 43240 total hands. This means you play a 3 card royal about every 66 hands.
