Blackjack Probability/Odds Formulas

I was reading the Wizard of Odds Web-Site and the tables they have published for specific hand probability of side bets. The link is located here:

I am getting thrown off by some of the calculations because the site does not include formulas. I am interested in the math used to determine the statistical probability of a specific hand being dealt.

For example, Wizard of Odds states that for a six deck game, the probabilities of getting Lucky Ladies (Pair of Queen of Hearts) is 0.0295%. I cannot determine how this was calculated. By my own math, which is clearly not accurate, I am calculating 6 x 5 / 48,516 to equal 0.062% probability of occurrence rate. I determined this as follows:

6 Queen of Hearts in the 312 card shoe.
Once the first is received, there are only 5 QH cards left that you may possibly be dealt.
312 times 311 divided by 2 = the quantity of possible 2-card combinations in the 312-card deck, 48,516.

What am I doing wrong in this example?

The reason I am asking this is because my friends and I are putting together a weekly game and we want to have a bunch of fun side bets. We are playing at a rate of $0.01 penny = 1 chip, so we aren't necessarily too concerned about the house edge being somewhat favorable to the player since we are all funding the house each month via monthly contributions. But, we also don't want to make the conditions so advantageous that the house is at a strong disadvantage. It was my thought process that since the side-bets favor the house, having many that are appealing would balance out other conditions that are more favorable than a Casino. Mostly we just want the side bets because they are fun. Here are the details of the rules we are contemplating using. I would really appreciate anyone taking the time to provide the math on probabilities (with formulas illustrating how you came to your conclusion), and suggest payout ratios, i.e. Lucky Ladies pays 125-1, and 1000-1 on Dealer Blackjack.

Thanks very much, hope you find some of these side bet ideas amusing.

House Rules: "Royal 21" Blackjack Variant
6 Decks (312 cards)
3 Jokers
1 Gaff Card (aka X-Card)
Total Cards: 316

Dealer hits on a soft 17:
Player can double after a split
Player can double on any first two cards
Player can re-split to 4 hands
Player can re-split Aces
Player can hit split aces
Surrender rule: None
No Insurance Offered except Even Money for player Blackjack
Blackjack pays: 3 to 2

Side Bets:

Kingdom Forged in Blood.....First four cards are Red Aces with 21 achieved on at least one hand
Black Knight's Conquest.......First four cards are Black Aces
Angel's Fury........................First three cards are suited Aces
The Devil's Banquet.............First three cards are Red 6's
Supremacy of the Crown......Four cards consisting of player's and dealer's first two cards are all suited face cards
Consequence of Sorcery.......First three cards are 7's of the same color
Conspiracy of the Jesters.....Receive two Jokers in the same hand (does not need to be first two cards)
King's Bounty......................First two Cards are King of Spades
Lucky Ladies.......................First two cards are Queen of Hearts
Jack Magic..........................First two cards plus dealer's up card are Jacks of any suit
Joker's Wild........................First two cards are Joker and Ace of any suit
Royal Match.......................First two cards are suited King and Queen
High Society.......................First two cards are face cards of any suit
Dealer's Downfall................Win if dealer busts, regardless of outcome of player's hand

Executioner's Malice............Gaff/X-Card dealt to player and hand equals 18 - 21. X-Card itself has no numeric value. Player may opt to ignore X Card and play hand as normal. Player is ineligible to use X-Card if standing on 17 or below or if hand busts. If player receives hand valued 18 - 21, player can force dealer and all other players at the table to bust, including against a dealer Blackjack. Player will receive payout for primary bet, collect all other player's primary bet, and receive payout on side bet of TBD value.

The .0295% is for a pair of Qh without the dealer having a Blackjack, the probability of a pair of Qh AND the dealer having a Blackjack is .0015%. The combined probability (any hand with a pair of Qh) is:

(6/312 * 5/311) = 0.00030917635

For all of your list of side bets, I'll do the easy ones really fast, and I'm also doing these all based on standard six decks. If you want it with your decks, someone else will have to come along, or you can get the idea how to do it from these.

Kingdom Forged in Blood

You would need at least three players and a dealer for this to work correctly, otherwise you have less than four hands which reduces the probability of someone getting a blackjack, unless you are counting splits for, "21's," rather than Blackjacks.

(12/312 * 11/311 * 10/310 * 9/309) = 0.00000127814

Okay, now at least one of these hands has to be a Blackjack, so it's easier to calculate the probability of NONE of them being a Blackjack and subtracting that from one.

1-(212/308 * 211/307 * 210/306 * 209/305) = 0.77752893222

Okay, so you take the probability of the four consecutive red aces and multiply it by the probability of one of the hands making a Blackjack afterward:

0.00000127814 * 0.77752893222 = 9.93790829e-7 or 0.000000993790829 or 1/0.000000993790829 = 1 in 1006247.96569

In other words, it'll be awhile before you have to worry about that happening. Even four consecutive colored Aces and a total of 21 (BJ or otherwise) is quite unlikely.

Black Knight's Conquest

(12/312 * 11/311 * 10/310 * 9/309) = 0.00000127814 or 1/0.00000127814 = 1 in 782386.905973

Angel's Fury

(24/312 * 5/311 * 4/310) = 0.00001595748 or 1/0.00001595748 = 1 in 62666.5363203

The Devil's Banquet

(12/312 * 11/311 * 10/310 * 9/309) = 0.00000127814 or 1/0.00000127814 = 1 in 782386.905973

Supremacy of the Crown

(72/312 * 17/311 * 16/310 * 15/309) = 0.00003160512 or 1/0.00003160512 = 1 in 31640.4430675

Consequence of Sorcery

(12/312 * 11/311 * 10/310 * 9/309) = 0.00000127814 or 1/0.00000127814 = 1 in 782386.905973

King's Bounty

(6/312 * 5/311) = 0.00030917635 or 1/0.00030917635 = 1 in 3234.40004386

Lucky Ladies

See Above.

Jack Magic

(24/312 * 23/311 * 22/310) = 0.00040372447 or 1/0.00040372447 = 2476.93680792

Royal Match

(48/312 * 6/311) = 0.002968093 or 1/0.002968093 = 1 in 336.916666695

High Society

(72/312 * 71/311) = 0.05268365075 or 1/0.05268365075 = 1 in 18.9812206589

Dealer's Downfall

The probabilities of a dealer bust for any number of decks can be found here:

https://wizardofodds.com/games/blackjack/appendix/2b/

And are a little better than 1 in 4.

Summary

Those are the probabilities of the events based on six standard decks and no lower pay other than the ones mentioned. As you can see, some of these would have to pay hundreds, even thousands of dollars, on a $.01 bet, not to have a ridiculously high house edge unless combined with something else. For example, if your Black Knight's Conquest paid $5,000, then:

(5000 * 0.00000127814) - ((.01 * (1-0.00000127814)) = -0.00360928721 which is a 36.0928721% House Edge.

Mission146, I am very grateful and appreciative of the time you took to respond and provide this valuable feedback. The examples you provided will certainly enable me to do some # crunching on my own. I also appreciate the insight on payout levels necessary to keep house edge in check. Hopefully we can build a nice bank over time to start paying some serious cash on the side bets. You are a wealth of insight and knowledge, thank you so much, I hope you have a wonderful holiday season in the weeks ahead!

PS: Earlier this year I witnessed a woman playing at my table get 4 red aces and nail a blackjack on the first hand and a 21-count on the third split hand, which is how I thought of that one. I was just referring to 21-count, not necessarily blackjack, but very interesting to see the odds you provided. Thanks again!

Quote: JGott

Mission146, I am very grateful and appreciative of the time you took to respond and provide this valuable feedback. The examples you provided will certainly enable me to do some # crunching on my own. I also appreciate the insight on payout levels necessary to keep house edge in check. Hopefully we can build a nice bank over time to start paying some serious cash on the side bets. You are a wealth of insight and knowledge, thank you so much, I hope you have a wonderful holiday season in the weeks ahead!

PS: Earlier this year I witnessed a woman playing at my table get 4 red aces and nail a blackjack on the first hand and a 21-count on the third split hand, which is how I thought of that one. I was just referring to 21-count, not necessarily blackjack, but very interesting to see the odds you provided. Thanks again!

I screwed up Devil's Banquet, you said the first three cards are Red 6's, not the first four. In any case, the probability I did for that is the same as the probability of the first four cards being Red Aces, so, around 1/750,000.

I hope you have a wonderful holiday season, as well, have fun with your Blackjack games. I should also say that The Wizard is a wealth of knowledge, not me, he'd put together a program to analyze some of those tougher side bets.

I definitely like some of the names, by the way, I don't know that the names would take off in a casino...but they are creative!

Quote: Mission146

I definitely like some of the names, by the way, I don't know that the names would take off in a casino...but they are creative!

Thanks, maybe only at Excalibur, ha! If you have better ideas, I'm all ears, whether suggestions for better names or suggestions for alternative side bets that you think would be fun, trying to flesh out this " Royal 21" variant rules.

I tried to associate the names with the cards dealt and royal/medeival nature of a traditional deck and existing side bet names. Yeah 666, three red sixes, that is the devil's hand for sure. Getting such a victorious hand of all red Aces, thats just total conquest, but with the red color thought I'd tie it to a bloody battlefield concept. Three 7s, seems impossible so if it happens, I think we can all agree that is a consequence of sorcery and not statistical mathematics, has to be...

So given that with these variant rules a player would be eligible for all of these side bets by placing a single side bet, and some are fairly common like dealer busting, isn't it collectively somewhat more fair than a casino, who typically only offer two or three side bets? My friends and my goal is that in a game of six or seven players, playing for 2 hours or so at a time, that at least one of those players would benefit from side betting to keep the allure and enticement of the side bet alive, while the majority would lose their side bets, and overall side betting is profitable for the house. Your feedback makes me think I need to add a few easier to achieve side bets in order to realize these objectives.

Thank you!

Quote: JGott

Thanks, maybe only at Excalibur, ha! If you have better ideas, I'm all ears, whether suggestions for better names or suggestions for alternative side bets that you think would be fun, trying to flesh out this " Royal 21" variant rules.

I tried to associate the names with the cards dealt and royal/medeival nature of a traditional deck and existing side bet names. Yeah 666, three red sixes, that is the devil's hand for sure. Getting such a victorious hand of all red Aces, thats just total conquest, but with the red color thought I'd tie it to a bloody battlefield concept. Three 7s, seems impossible so if it happens, I think we can all agree that is a consequence of sorcery and not statistical mathematics, has to be...

So given that with these variant rules a player would be eligible for all of these side bets by placing a single side bet, and some are fairly common like dealer busting, isn't it collectively somewhat more fair than a casino, who typically only offer two or three side bets? My friends and my goal is that in a game of six or seven players, playing for 2 hours or so at a time, that at least one of those players would benefit from side betting to keep the allure and enticement of the side bet alive, while the majority would lose their side bets, and overall side betting is profitable for the house. Your feedback makes me think I need to add a few easier to achieve side bets in order to realize these objectives.

Thank you!

JGott,

Your sidebet sounds like fun (though complicated - which I like, but is generally a drawback); however, I think (I'm not a math guy) you'll have to take an additional step and form all your bets into an odds paytable that, collectively, won't create a player edge. However, with so many options to win, your paytable could be designed to reflect virtually any house edge you want to offer. As far as fair, well, that will depend on what HE you offer the players, especially since you're kind of conflicted with all of you owning the house. I think a fair sidebet offers a fun paytable (at least one high-odds payout) with an overall HE below 5%. But that's me, usually betting $5-10/hand on a sidebet. And with a minbet of .01, you might want to set the HE between 10-15%, maybe higher, just to build any decent house bank. JMHO

Well, thanks to everyone's help we were able to put together our rules with a probability table, and we are having a blast. Thought I'd share our variant game, pictured below. First game paid heavily to the players, second game the house came out far ahead, so we'll just have to keep playing to determine fairness in the long run. Thanks again for the help.

Is this game in a casino or being developed for casino play? How many hands per hour would be expected?

Why is the side bet max only $10? Do you think any of these bets could be counted or otherwise exploited? The side bet names are very catchy, brilliant even, and would no doubt spice up the game. Imagine a table of imbibing players at 3AM. :-)

Hi 1BB. No, it's just a game being developed for my living room to play with friends. I'm not the greatest dealer so the speed and # of hands per hour, that is probably around 60 - 100, not sure I am not counting. I am getting better.

Not sure if the side bets can be counted or exploited. Probably the first three with the highest probability. I personally am not concerned because my friends and I don't count, and they are drinking as well. For this game to be played against a completely sober and experienced card counter, I have no idea. Would love to hear others thoughts.

The first two side bets together have a probability of being hit 1 in 8 times.The first three side bets together have a probability of being hit 1 in 7.7 times. The third bet pays 2-1 on blackjack versus normal 3-2, so that already gives the player more than they would normally earn. Therefore we have decided to limit the side bet amount. If there is some defects in our reasoning, by all means call it out here. But so far, we've felt the payouts have been appropriate and have had a lot of fun.

Thanks for the compliment on the side bet names, my friends and I think they add a bit of fun! We tried to be creative. We are also playing with Bicycle Tragic Royalty cards that glow in the dark under UV light, so that adds a pretty awesome factor as well. No need to imagine the imbibing players at 3 AM, that was us 8 hours ago! :-)

Quote: JGott

we want to have a bunch of fun side bets.

I'm not sure I understand - is it 1 sidebet with a combination of 18 payout schedules, or 18 separate sidebets?

In either case, I think it's not going to find a place in a casino (except, possibly, at an electronic table or hybrid (iTable)) - just too complicated. (Executioner's Malice - no way it'll work in a casino, in my opinion. One of the principles of casino 21 games is that you play against the dealer, not the other players.)

(I already find myself having to remind dealers to pay other player's 21+3 bets, and that's 1 bet with like 5 ways to pay.)

It sounds like you've modified the KB/LL bets to be just the KS/QH pairs, instead of any 20 total. This makes them less vulnerable.

While I have great faith in "the percentage", I worry about your payouts that are over 1000:1. I understand that it's unlikely to hit, but someone betting 20 cents (minimum on the spot, max on the sidebet) has a potential to win $1000 if the Kingdom Forged in Blood hits. Since I'm not sure of the mechanics of your sidebets, I don't know if all 7 player spots could hit it on the same hand. I'm sure you've got the house bank all squared away and more than ready to settle any such outcome, and any such concerns are wholly without merit. (Right? You've got $10k or more on hand, just in case, at a friendly game for pennies?)

I would suggest having the rules for each carefully written out, unambiguously, and unanimously agreed upon. Don't assume that everyone just knows how it works - there's too much potential for hard feelings.

A lot of these are interesting, and highly amusing.

Hi Dieter, thanks for your input, it is very appreciated.

It is one side bet eligible for the combination of all 18 possible payout scenarios.

Man that is crazy that you've been in situations where you have to remind the dealer to pay out a side bet. That's just wrong. But yes, obviously our long list is complicated due to the quantity and varying rules of each, so certainly it is only going to work well when playing it routinely with the same group of people who all understand the details of each. The most complicated ours get is a 3 card combination, so it isn't too mind-boggling. But agreed, this would prove disastrous if offered in a setting where people unfamiliar with them all just sat down and started playing, that would be very disruptive and slow down play. I think one or two of our brain-children here could work in a Casino setting, as long as the side bets were limited to only 2 or 3 options.

I agree 100% about our Executioner's Malice Side Bet, a side bet that impacts other players would not work at a Casino, a player should play against the dealer and not impact the other players in any way. It would certainly be foolish for a Casino to introduce such a side bet. However, when playing with just friends, there is a satisfaction of taking all of your friends money and forcing them to bust!

Yes, the King's Bounty and Lucky Ladies removed all other facets of the bet some casinos may offer, it is only KS/QH pairs. But separately, we have introduced High Society, which is basically the same as the two cards equal 20 bet that is normally part of Lucky Ladies at the Casino, except our version just removes the "10" card and is only winnable if the player gets a J,Q,K. That is still a 1 in 20 chance so that is pretty good, especially since it pays 5 - 1. If you are smart and are counting the times that is occurring you could stagnate your side betting and perhaps come out ahead. My friends are too drunk and distracted to accomplish this feat but perhaps it could cost me when playing with more attentive individuals.

You are right about the payout concerns, if we were playing with full dollars, certainly we would fund the bank with $20K or so to ensure payout coverage. But we are playing with pennies. Each one of our $1 chips is just a penny. That being the case, we realize that some lucky person who gets the Kingdom Forged in Blood side bet would win $1,000 if they had a maximum side bet of 10 (in our case this is a $0.10 side bet). "The bank" is adequately funded to cover that unlikely scenario. Given that it is unlikely that it would occur more than once in 550,000 times (which considering 2-3 hours of play once every two weeks, approx 1000 hands per week, it would take 20 years to play that many hands), honestly it would be worth the money to pay it out just for our own amusement, even if it did cause a loss. But, if someone were to receive that lucky hand, it is also more probable that they will not have a $0.10 side bet. It is more probable they will only have a $0.01 side bet, which would pay only $100, or no side bet at all.

To your question if all players could hit a side bet on the same hand, yes multiple players could hit the same side bet on the same round of dealing, if there are enough cards. For the highest paying one, Kingdom Forged in Blood which requires three Red Aces and a blackjack on at least one of the three hands, the formula is (6/160)*(5/159)*(4/158)*(0.0601415) (the blackjack probabilities are higher because of the three Jokers). So yes, if we were playing in the Bermuda Triangle, two players could receive that 1 in 550,000 chance side bet on the same hand.

Some other hands that are more probable, like High Society (two face cards), we've already witnessed that happening to two players on the same round of dealing.

Good suggestion about the rules for each side bet being carefully written out in more detail. We will do that and make a booklet. Thank you very much!

Quote: JGott

if we were playing with full dollars, certainly we would fund the bank with $20K or so to ensure payout coverage. But we are playing with pennies.

If you were playing with dollars, a max sidebet Kingdom Forged in Blood hit would cost you $100,000, and you'd still be woefully underfunded. (Fundamental principle: don't overbet your bankroll, even when you have an edge.)

I'm glad you can cover at least one hit on it.



Completely unrelated question - what's your house shuffle look like?

We are practicing a shuffle at the 2-deck penetration point, or upon the second Joker being dealt, whichever comes first.

Quote: JGott

We are practicing a shuffle at the 2-deck penetration point, or upon the second Joker being dealt, whichever comes first.

You're only dealing 2 decks? That's spending half your time shuffling instead of playing.

Perhaps you deal 4 decks, or to the second joker?


I was actually more curious about the way you shuffle the cards, instead of when you shuffle the cards, but I understand if you're protecting your high stakes game by not discussing it. ;)