Bovada BJ Expectation

Bovada 6 deck BJ expectation -


Would someone be willing to check my figures and see if I'm correct?
Thx in advance.


Bovada 6 Deck Blackjack -

Rules (copied from Bovada site) -


The game is played using six decks, which are reshuffled after each hand.
Dealer hits on soft 17.
Player Blackjack pays 3 to 2.
Any other winning hand is paid 1 to 1.
Insurance pays 2 to 1.
Split up to three hands.
Surrender any first two cards.
Split aces receive only one card.
No re-splitting of aces.
A split ace and a ten-value card is not a Blackjack.

From The Wizard:


Blackjack House Edge Calculator

Enter any set of blackjack rules from the options below. The house edge under proper basic strategy for these rules is indicated in the box below.


Number of decks of cards used: 1 2 4 5 6 8 - (6 decks in my case)
Dealer hits or stands on a soft 17: Stands Hits - (Hits)
Player can double after a split: No Yes - (Yes)
Player can double on: Any first two cards 9-11 only 10-11 only - (Any first two cards)
Player can resplit to: 2 3 4 hands - (3 hands)
Player can resplit aces: No Yes - (No)
Player can hit split aces: No Yes - (No)
Player loses only original bet against dealer BJ: No Yes - (Yes)
Surrender rule: None Late - (Late)
Blackjack pays: 3 to 2 6 to 5 - (3:2)

Optimal results: (0.53594%, 0.0053594)

Basic strategy with cut card: (0.55904%, 0.0055904)

Basic strategy with continuous shuffler: (0.53904%, 0.0053904)

Methodology: The "optimal results" are based on perfect composition dependent strategy (composition dependent strategy advantage negligible on more than 1 deck) and the dealer shuffling after every hand, which benefits the player. The "basic strategy with cut card" results are based on total dependent basic strategy, like the tables on this site, and the use of a cut card, which favors the dealer. The "basic strategy with continuous shuffler" results are based on total dependent basic strategy, and the dealer shuffling after every hand.


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


I've copied his advanced composition dependent strategy and use it religiously on single deck games.




Here's what I get:


SD confidence intervals -

1 SD, ~ 68%
2 SD's, ~95%
3 SD's, ~99.7%
4 SD's, ~99.993%
5 SD's, ~99.99994%

I read somewhere The Wiz says he wouldn't suspect a casino unless players were getting frequent returns at or below (-4 SD's + HE) and wouldn't report a casino as dishonest unless losses occurred frequently at or below (-5 SD's + HE).


D - deposit ($100)
B - bonus ($100)
R - rollover rate (40X)
b - (flat) bet ($1)
RO - rollover = RO*(D+B) = 40*($200) = $8 000
N - number of hands, N = RO/b = $8 000/$1 = 8 000 hands


SD(bj) - BJ generic SD (~1.15)

SD(orig) - original SD

SD(orig) = SD(BJ)*b = 1.15*$1.00 = $1.15

SD(N) - SD applicable to N hands

SD(N) = SD(orig)*sqrt (N)

= b*SD(bj)*[RO/b]^(1/2)

= [b^(1/2)]*SD(bj)*RO^(1/2)

= [$1^(1/2)]*1.15*$8 000^(1/2)]

= $102.86


1 SD ~$102.86, [-$50.88, $154.84], +- $51.98 ~68% confidence
2 SD ~$205.72 [-$153.74, $257.70], +- $51.98 ~95% confidence
3 SD ~$308.58 [-$256.60, $360.55], +- $51.98 ~99.7% confidence



HE - House Edge = (0.0053904, 0.53904%), per Wiz


BDCP - Bovada Deposit Charge Percentage = (0.0049, 0.049%)

BDC - Bovada Deposit Charge = BDCP*D = 0.049*$100.00 = $4.90

BDC(RO) - Bovada Deposit Charge over the Rollover = BDC/RO = $4.9/$8 000 = (0.0006125, 0.06125%)


BE - Bonus Expectation = B/RO = $100/$8 000 = (0.0125, 1.25%)


E(T) - Total Expectation = HE + BDC(RO) + BE = -0.0053904 + 0.0125 - 0.0006125 =

(0.0064971, 0.64971%) > 0, a positive expectation game.

E(T,a) - Expectation, Total, amount = E(T)*RO = 0.0064971*$8 000 = $51.98



Finally -

Game has a consistent positive player expectation (0.63%, with deposit charge, bonus. Other charges not figured in).

Whilst I didn't re-work all the math, eyeballing it appears to be mostly correct. I think you do SD a bit different (not necessarily incorrectly though) than me as I usually have an expectation and the +-SD grows with each confidence interval, where as your +-SD remains the same and the expectation changes:

Quote: APEppink

1 SD ~$102.86, [-$50.88, $154.84], +- $51.98 ~68% confidence
2 SD ~$205.72 [-$153.74, $257.70], +- $51.98 ~95% confidence
3 SD ~$308.58 [-$256.60, $360.55], +- $51.98 ~99.7% confidence

Perhaps I missed the comma in your quote above... It appeared as though your 1SD, 2SD, 3SD were the expectation and the +-51.98 was your SD.

For your 8,000 hands... EV = TotalWagered * HE = NumBets * AvgBet * HE = (8000)*(1)*(-.0054) = -43.2
*I used the continuous shuffler HE because the rules state it's shuffled every hand.

OriginalSD = 1.15*AvgBet = 1.15*1 = $1.15
SD(8000) = OriginalSD * Sqrt(8000) = $102.86

So in 8k hands you could expect to lose EV +-SD:
-$43.20 +-$102.86 for 1SD... for a range of -$146.06 to $59.66
-$43.20 +-$205.72 for 2SD... for a range of -$248.92 to $162.52
-$43.20 +-$308.58 for 3SD... for a range of -$351.78 to $265.38

It is possible to get an edge still, but it's much lower than prior days of 'ol. My brother wanted to play in some donk poker tourneys with me and I opted in to a casino bonus and found if I ran 9/6 JoB I had a positive expectation on my bonus... if I survived the variance of the game.

Quote: Romes

Whilst I didn't re-work all the math, eyeballing it appears to be mostly correct. I think you do SD a bit different (not necessarily incorrectly though) than me as I usually have an expectation and the +-SD grows with each confidence interval, where as your +-SD remains the same and the expectation changes:

Quote: APEppink

1 SD ~$102.86, [-$50.88, $154.84], +- $51.98 ~68% confidence
2 SD ~$205.72 [-$153.74, $257.70], +- $51.98 ~95% confidence
3 SD ~$308.58 [-$256.60, $360.55], +- $51.98 ~99.7% confidence

Perhaps I missed the comma in your quote above... It appeared as though your 1SD, 2SD, 3SD were the expectation and the +-51.98 was your SD.

For your 8,000 hands... EV = TotalWagered * HE = NumBets * AvgBet * HE = (8000)*(1)*(-.0054) = -43.2
*I used the continuous shuffler HE because the rules state it's shuffled every hand.

OriginalSD = 1.15*AvgBet = 1.15*1 = $1.15
SD(8000) = OriginalSD * Sqrt(8000) = $102.86

So in 8k hands you could expect to lose EV +-SD:
-$43.20 +-$102.86 for 1SD... for a range of -$146.06 to $59.66
-$43.20 +-$205.72 for 2SD... for a range of -$248.92 to $162.52
-$43.20 +-$308.58 for 3SD... for a range of -$351.78 to $265.38

It is possible to get an edge still, but it's much lower than prior days of 'ol. My brother wanted to play in some donk poker tourneys with me and I opted in to a casino bonus and found if I ran 9/6 JoB I had a positive expectation on my bonus... if I survived the variance of the game.

Thx for replying. You're right. I fouled up the SD signs. It should be:

1 SD ~+-$102.86, [-$50.88, $154.84], +$51.98 ~68% confidence
2 SD ~+-$205.72, [-$153.74, $257.70], +$51.98 ~95% confidence
3 SD ~+-$308.58, [-$256.60, $360.55], +$51.98 ~99.7% confidence

Where we differ is you used the Wiz' house expectation, while I figured in not only the Wiz' expectation but the effects of the Bovada deposit charge (4.9%) and their matching bonus ($100) as well, for a positive expectation of (0.0064971, 0.64971%), (see below). I think it's correct, at least I hope so:


HE - House Edge = (0.0053904, 0.53904%), per Wiz


BDCP - Bovada Deposit Charge Percentage = (0.0049, 0.049%)

BDC - Bovada Deposit Charge = BDCP*D = 0.049*$100.00 = $4.90

BDC(RO) - Bovada Deposit Charge over the Rollover = BDC/RO = $4.9/$8 000 = (0.0006125, 0.06125%)


BE - Bonus Expectation = B/RO = $100/$8 000 = (0.0125, 1.25%)


E(T) - Total Expectation = HE + BDC(RO) + BE = -0.0053904 + 0.0125 - 0.0006125 =

(0.0064971, 0.64971%) > 0, a positive expectation game.

E(T,a) - Expectation, Total, amount = E(T)*RO = 0.0064971*$8 000 = $51.98


Re variance. You're right. It'll eat you up if you're not careful (e.g. Full Pay Deuce's Wild etc.).


May I ask what your background is? Mine's Mechanical Engineering (BSME).

Thx. again for your reply.

Yeah I did not account for the Deposit Charge, nor go as far as to produce an overall edge. While we were dollars off for these reasons in again eyeballing your math does appear correct. I haven't checked yet, but also you might want to check if you can do the play through requirements on blackjack. Some sites don't let you. I was going to actually ask bovada if I could do that as well and do something similar =).

I lived with a couple ME's =P. I was CEG/CS with a minor in Mathematics.

If you're willing to play with such a low edge and grind you should deposit the maximum and take advantage of all the bonuses including the 50% sports deal.

Then do some weekly bonuses (play it during their drawing if they still have them)hopefully that will help dilute the deposit fee.

Sometimes they give you a bonus for hitting your first blackjack on certain days IIRC it was on Sunday's or holidays ($5 or $10 min bet).

Always check on holidays that's when the used to run special deals.