9/5 Blackjack

BetUS has a promotion on Thursdays where a Blackjack pays 9/5, instead of 3/2 or the dreaded 6/5. I've looked and can't find any information on what this does to the house edge. I know that 6/5 increases the house edge on average from 0.45% to 1.82%.

As 9/5 is better than 3/2, does anyone know how to compute the improved edge?

Quote: Local871

BetUS has a promotion on Thursdays where a Blackjack pays 9/5, instead of 3/2 or the dreaded 6/5. I've looked and can't find any information on what this does to the house edge. I know that 6/5 increases the house edge on average from 0.45% to 1.82%.

As 9/5 is better than 3/2, does anyone know how to compute the improved edge?

This is actually a simpler problem than you would think. The first thing that we want to do is look at this:

https://wizardofodds.com/games/blackjack/calculator/

I'm just leaving the rules that are already in there the same. 6L5 has a House Edge of 1.78895% and 3:2 is 0.43096%, so:

1.78895% - 0.43096% = 1.35799% is the difference.

With that, we're going to take a trip back in time and concern ourselves with the Least Common Denominator, that way, we will index 6/5 and 3/2 to being the same thing. Let's go ahead and do that:

We start with the fact that our Least Common Denominator is 10, so that makes the adjusted fractions thus:

15/10 (3:2) and 12/10 (6/5)

The next thing we do is ask ourselves, "What would this adjusted fraction be for 9:5 Blackjack?" That's easy: It is 18/10.

We see that 1.35799% is the difference between 15/10 (3:2) and 12/10 (6:5) Blackjack, and relative to the Least Common Denominator of 10, the difference in our numerator (top number) is three.

The next thing we posit is that the probability of a winning player natural does not change, so any changes in the pays are unaffected by strategy.

Our 18:10 Blackjack goes three up in the numerator compared to the base game instead of three down, as a result, 1.35799% is subtracted from the House Edge.

0.43096 - 1.35799 = -0.92703 or 0.92703% Player Advantage

As we can see, it's the same thing as making it 6:5 Blackjack, just the other way.

1.) This method would NOT work for most strategy-dependent casino games. The only reason it works here is because the probability of a winning player natural cannot be changed by strategy. This would work for adjusting pays on side bets, for instance, provided there are no strategy changes that would make a particular result more or less likely. For most side bets, there are no such changes as many of them are just based on whatever it is you are dealt.

2.) But, I do hope that this was a useful example of how to use what you already know to get to what you want to figure out.

3.) You also know now that, having indexed it to ten, every 1/10th unit pay up or down changes it 1.35799/3 = 0.45266333333...at least for the rules linked above. (Only number of decks will change this)

Quote: Mission146

This is actually a simpler problem than you would think. The first thing that we want to do is look at this:

https://wizardofodds.com/games/blackjack/calculator/

I'm just leaving the rules that are already in there the same. 6L5 has a House Edge of 1.78895% and 3:2 is 0.43096%, so:

1.78895% - 0.43096% = 1.35799% is the difference.

With that, we're going to take a trip back in time and concern ourselves with the Least Common Denominator, that way, we will index 6/5 and 3/2 to being the same thing. Let's go ahead and do that:

We start with the fact that our Least Common Denominator is 10, so that makes the adjusted fractions thus:

15/10 (3:2) and 12/10 (6/5)

The next thing we do is ask ourselves, "What would this adjusted fraction be for 9:5 Blackjack?" That's easy: It is 18/10.

We see that 1.35799% is the difference between 15/10 (3:2) and 12/10 (6:5) Blackjack, and relative to the Least Common Denominator of 10, the difference in our numerator (top number) is three.

The next thing we posit is that the probability of a winning player natural does not change, so any changes in the pays are unaffected by strategy.

Our 18:10 Blackjack goes three up in the numerator compared to the base game instead of three down, as a result, 1.35799% is subtracted from the House Edge.

0.43096 - 1.35799 = -0.92703 or 0.92703% Player Advantage

As we can see, it's the same thing as making it 6:5 Blackjack, just the other way.

1.) This method would NOT work for most strategy-dependent casino games. The only reason it works here is because the probability of a winning player natural cannot be changed by strategy. This would work for adjusting pays on side bets, for instance, provided there are no strategy changes that would make a particular result more or less likely. For most side bets, there are no such changes as many of them are just based on whatever it is you are dealt.

2.) But, I do hope that this was a useful example of how to use what you already know to get to what you want to figure out.

3.) You also know now that, having indexed it to ten, every 1/10th unit pay up or down changes it 1.35799/3 = 0.45266333333...at least for the rules linked above. (Only number of decks will change this)

I am not going to go thru all your calculations but a very reliable Blackjack software called MGP's Blackjack Combinatorial Analyzer gives the following for 6 decks, S17, DAS and BJ 3:2

House Edge is 0.404%

When modifying it to BJ 9:5, the House Edge goes negative to -0.956

So the difference between BJ 9/5 and BJ 3/2 is 1.36% IN FAVOR OF THE PLAYER.

Quote: GMan

I am not going to go thru all your calculations but a very reliable Blackjack software called MGP's Blackjack Combinatorial Analyzer gives the following for 6 decks, S17, DAS and BJ 3:2

House Edge is 0.404%

When modifying it to BJ 9:5, the House Edge goes negative to -0.956

So the difference between BJ 9/5 and BJ 3/2 is 1.36% IN FAVOR OF THE PLAYER.

Thank you very much for your post. The software agrees with what I have said. The slight difference in percentages is due to mine being a calculation based on eight decks.

Quote: GMan

I am not going to go thru all your calculations but a very reliable Blackjack software called MGP's Blackjack Combinatorial Analyzer gives the following for 6 decks, S17, DAS and BJ 3:2

House Edge is 0.404%

When modifying it to BJ 9:5, the House Edge goes negative to -0.956

So the difference between BJ 9/5 and BJ 3/2 is 1.36% IN FAVOR OF THE PLAYER.

That’s exactly what Mission146 math said. Far from not going through it you didn’t even read the conclusion.

Quote: unJon

That’s exactly what Mission146 math said. Far from not going through it you didn’t even read the conclusion.

Thanks for your post also!

It's probably irrelevant for him because the software gives the correct answer. I could have given the answer without typing out my calculations, but I perceived value in teaching the OP the underlying concept of how to figure this out for himself as similar math might be used for other problems.

Some people would just want an accurate answer, which is also fine.

Sounds like a math problem...

Quote: GMan

I am not going to go thru all your calculations but a very reliable Blackjack software called MGP's Blackjack Combinatorial Analyzer gives the following for 6 decks, S17, DAS and BJ 3:2

House Edge is 0.404%

When modifying it to BJ 9:5, the House Edge goes negative to -0.956

So the difference between BJ 9/5 and BJ 3/2 is 1.36% IN FAVOR OF THE PLAYER.

You do not need any software to calculate this edge. The player gains an additional edge of 4.7%x(1-4.7%)x(9/5-3/2)=1.3%.

Quote: ThatDonGuy

Sounds like a math problem...

Challenge Accepted!

What is the probability of getting a blackjack from a full shoe of D decks?

There are 52 D cards, of which 4 D are Aces and 16 D are 10-value, so of the 52 D x (52 D - 1) / 2 = (1352 D^2 - 26 D) combinations of two cards, of which 4 D x 16 D = 64 D^2 are blackjacks, so the probability of being dealt a blackjack is 64 D^2 / (1352 D^2 - 26 D) = 32 D / (676 D - 13)

The gain per blackjack from 3-2 is 3/10, so the added player advantage is 32 D / (676 D - 13) x 3 / 10 = 96 D / (6760 D - 130) = 48 D / (3380 D - 65)

However, you have to take into account that the dealer does not also have a blackjack.
With D decks, there are now (52 D - 2)(52 D - 3) / 2 = 1352 D^2 - 130 D + 3 combinations of cards remaining, of which (4D - 1)(16 D - 1) = 64 D^2 - 20 D + 1 blackjacks, so the probability of the dealer having a blackjack is (64 D^2 - 20 D + 1) / (1352 D^2 - 130 D + 3), and the probability that the dealer does not is (1288 D^2 - 110 D + 2) / (1352 D^2 - 130 D + 3).

Thus, the added player advantage is 48 D / (3380 D - 65) x (1288 D^2 - 110 D + 2) / (1352 D^2 - 130 D + 3).

For 1 to 8 decks in the shoe:
1: 1.394773% (3,776 / 270,725)
2: 1.373499% (157,888 / 11,495,315)
3: 1.366558% (16,384 / 1,198,925)
4: 1.363115% (258,176 / 18,940,155)
5: 1.361057% (506,432 / 37,208,717)
6: 1.35969% (877,632 / 64,546,495)
7: 1.358714% (6,982,528 / 513,906,965)
8: 1.357984% (10,438,912 / 768,706,575)

Very nice! I have hereby been appropriately humbled.