# Blackjack statistics

#### lucky21

##### Dormant account
Hi All
Can anyone give me a lay mans idea of the win loss stats of BJ, I have looked a lot at wizard of odds and understand the approx 48% loss 47% win and 5% push.

I am just wondering if anyone can tell me over how many hands played would you expect the above percentages to be seen, for example if I play 1000hands and only win 30% playing with basic strat, then it could be said it is to small a sample.

So how many hands should i play to get a reasonable sample to indicate if the game is playing correctly? I have asked this question of a gaming commision in malta and they dont seem to understand that if the statistics arent within the expected win loss or are well out as with the 30% win over how ever many hands is considered a fair sample, then something could be wrong with the game.

I know one thing for sure, that online bj continually gives large losses in a row that would seem impossible according to wizard of odds tables.

#### Benke

##### Dormant account
Its not a direct answer to your question but you can calculate the standard deviation, see
for more information. Anything above 4 or in the high 3 and its really unlucky or something is fishy with the software.

That link also indicates that the probability of a push is 8.48%, win 42.42% and loss 49.10%. But the wins are often more than one bet unit due to doubles and splits.

#### 3Dice

##### I-Gaming Industry Representative
Hey lucky, benke,

The standard deviation is only part of the answer .. to answer the question of the sample size conclusively would require a lot more mathematics than you would think at first.

Assuming a couple of things, like that we use random sampling, and that the frame is a good representation of the population, and so on and so on, we can simplify things a bit ..

The more samples you take, the more certain you get. So the question becomes .. how certain do you want to be. This certainty is usually called the 'standard error' and calculated as such :

standard error = standard deviation / square root ( number of samples )

so, the closer that number gets to 0, the more certain you are. E.g. 0.05 = 95% certain that the sample frame represents the population.

Practically, you write down each result, calculate the average at that point, calculate the standard deviation, then calculate the standard error untill its small enough for you to be 'certain'.

Hope that helps ..

Enzo@3Dice

#### lucky21

##### Dormant account
thanks

thanks dice for your info but at the risk of sounding lazy can you spell your process out for me, using a hypothetical number of hands wins loses, step by step so to speak.
Lucky

#### Random#rs

##### Dormant account
Yes they're correct its more complicated than you think. You can get an idea of how fair the game is in about a 1000 hands. It involves the standard of deviation as mentioned.

Figuring a flat base unit bet and using a perfect stategy, you have about a 10% chance of losing 47 bet units in 1000 hands. Thats about 10X the house edge. This would happen one in ten 1000-hand sessions.

If you stay below this figure your probably OK. Use the expected loss table to do this experiment for any number of hands but remember the fewer hands you run for a test, the less accurate it will be; but you can still get a good feel with a smaller test.

Use the Probability of Loss Table here at the bottom of page here:

To get a 100% chance of a House edge loss would require more hands than you would care to play.

Also, watch your Variance and see if it looks reasonably consistant:

According to Professional Blackjack by Stanford Wong (page 203), the variance for similar rules is 1.32 and the covariance is 0.48. The total variance of n hands would be 1.32*n + 0.48*n*(n-1). Take the final square root to get the standard deviation.

#### EasyRhino

##### Dormant account
All of this assumes you're playing perfect basic strategy, by the way.

#### Random#rs

##### Dormant account
According to Professional Blackjack by Stanford Wong (page 203), the variance for similar rules is 1.32 and the covariance is 0.48. The total variance of n hands would be 1.32*n + 0.48*n*(n-1). Take the final square root to get the standard deviation.
OK I didn't get a chance to finish my last post so I'm back and sorry about misspelling Perfect basic Strategy. Yes it has to be a fixed system of play so you have to use perfect basic strategy through the entire test. Also it must be a fixed bet size on every hand. Of course doubling and splitting are valid in the test.

The above formula gives us the standard deviation for Blackjack in any number of hands(n) . That means multi-hand not hands played. So you would figure it in probably 1 to 3 for the value n. It will be about
1.15514 for 1 hand BJ
1.34942 for 2 hand
1.51957 for 3 hand

To figure your games current standard of deviation you would have to keep track of the result of every hand played as x, (x= your current balance). You would have to do a number of calculations, too many; so get a math program for the PC that is capable of calculating standard of deviation for any series of numbers input, and enter your balance after each hand. It will do it all for you instantly. Optimally, your looking for 1.155 for single hand BJ but it will take a while before you see the result smooth out to something useful. You need enough data (hands played) to show a meaningful standard of deviation.

#### aka23

##### Dormant account
Hi All
Can anyone give me a lay mans idea of the win loss stats of BJ, I have looked a lot at wizard of odds and understand the approx 48% loss 47% win and 5% push.
Not sure where you got those numbers, but win % should be much lower... The numbers Benke posted are accurate. This may explain much or your results.

I am just wondering if anyone can tell me over how many hands played would you expect the above percentages to be seen, for example if I play 1000hands and only win 30% playing with basic strat, then it could be said it is to small a sample.

So how many hands should i play to get a reasonable sample to indicate if the game is playing correctly? I have asked this question of a gaming commision in malta and they dont seem to understand that if the statistics arent within the expected win loss or are well out as with the 30% win over how ever many hands is considered a fair sample, then something could be wrong with the game.

I know one thing for sure, that online bj continually gives large losses in a row that would seem impossible according to wizard of odds tables.
A simple way to check if results are within normal expectations is to use the blackjack return/variance calc on my site at www.beatingbonuses.com/calc.htm . It outputs 1 and 2 standard deviation ranges of return. To get more precise results, you can program in a specific house edge and standard deviation to correspond to the rules of the particular blackjack game.

To check a specific variable, such as win %, you can find some other stat calcs on the web using Google or use Excel. The calcs should allow you to program in an expected percentage of the variable, how many trials, and how many times the variable occurred.

On the page at
, wizard of odds gives some other ideas of how to prove a blackjack game to be unfair.

#### Random#rs

##### Dormant account
Good stuff Aka-, I'll have to check the site out more thoroughly. I was only trying to figure out how to enter the ' Wager ' parameter in terms of hands played with a fixed bet. So if I wasn't using a Bonus, and I wanted to play 1000 hands, it didn't seem correct to enter 1000 for wagering since doubling and splitting in BJ will increase the total wagered by an indeterminate amount. Is there a way to enter total hands played instead of total wagered?

#### aka23

##### Dormant account
Good stuff Aka-, I'll have to check the site out more thoroughly. I was only trying to figure out how to enter the ' Wager ' parameter in terms of hands played with a fixed bet. So if I wasn't using a Bonus, and I wanted to play 1000 hands, it didn't seem correct to enter 1000 for wagering since doubling and splitting in BJ will increase the total wagered by an indeterminate amount. Is there a way to enter total hands played instead of total wagered?
The relation between amount wagered and hands played is a determinate amount, which varies slightly with a particular set of rules. The ratio is about 1.13 with typical BJ rules. With 1000 hands, you can assume wagering of ~1.13 x 1000.

#### gfkostas

##### Ex-Bonus Whore
And one question i have always been thinking

Do you stand on a 3+ cards that make 16 against 10/Ace?

#### aka23

##### Dormant account
And one question i have always been thinking

Do you stand on a 3+ cards that make 16 against 10/Ace?
Under typical online blackjack rules, optimal strategy is as follows:

Hard 16 vs 10 -- Surrender if allowed, otherwise Hit on 2 cards, Stand on 3+ cards
Hard 16 vs A -- Surrender if allowed, otherwise Hit

#### happygobrokey

##### Dormant account
the wiz has compiled some "composition-dependant" strategies that shave minute amounts off the house edge, because the cards in one's hand can, in a select few circumstances, change the probabilities. mostly this is at lower numbers of decks since the fewer cards to begin with, the more each card you see matters. "whatever's in those cases is not in yours." but the strat cards in popular use (for bj) are ones that are merely total-dependant, where you follow the action in the square regardless of how many/which cards make that total up. you can find the composition-dependant strategies easily at the wiz's site. in my experience, you must hit a 3-card 16 all day long, however the more cards in my hand (say 6 or 7) i would be wanting to do so less and less since all the little ones just came out and surely a tenspot is in that deck somewhere. pontoon is where the number of cards in the hand matters, and then it's a whole new ball game. but for normal bj, i think hitting 16 is the way to go.

#### MPB

##### Dormant account
but for normal bj, i think hitting 16 is the way to go.
Why would you think that?
Like aka23 already said, default optimal strategy for a standrad multideck BJ obv. tells you the opposite for 3+ cards.
It's probably very close, anyway this default strategy is the well-known and mathematically undisputed optimum.

#### aka23

##### Dormant account
...you must hit a 3-card 16 all day long, however the more cards in my hand (say 6 or 7) i would be wanting to do so less and less since all the little ones just came out and surely a tenspot is in that deck somewhere. pontoon is where the number of cards in the hand matters, and then it's a whole new ball game. but for normal bj, i think hitting 16 is the way to go.
Under the vast majority of rule sets, standing on 16 vs dealer 10 has a higher EV than hitting with 3+ cards (but not with 2 cards). This is true across all numbers of decks. Computer sims have come to this conclusion, and it is near universally accepted among experts, including Wizard of Odds.

16 vs 10 is a very close call, where hitting and standing have a similar EV. If your hand is composed of 3+ cards, then you likely don't have a 10 in your hand and do have small cards in your hand. So when you have 3+ cards, there is an increased chance of drawing a 10 on your 16 and busting, and a decreased chance of drawing a small card on your 16 and winning the hand. The difference is not much but it is enough to change the optimal strategy for this close call hand from hitting to standing when you have 3+ cards.

#### happygobrokey

##### Dormant account
oh come on, live a little!

#### EasyRhino

##### Dormant account
oh come on, live a little!
THE MATH DOES NOT LIE.

:notworthy

But the fact that one measly extra card drawn, even out of an 8 deck shoe, can shift the play decision, does a really great job of illustrating how dang close the call is.

So, while it would be difficult to "feel" the difference in play strategy in multihand 16s... in any reasonable acheivable amount of regular play for one person, it still doesn't mean there's not a Right answer.

#### NASHVEGAS

##### Banned User - flamming, disrespecting admin,
THE MATH DOES NOT LIE.

:notworthy

But the fact that one measly extra card drawn, even out of an 8 deck shoe, can shift the play decision, does a really great job of illustrating how dang close the call is.

So, while it would be difficult to "feel" the difference in play strategy in multihand 16s... in any reasonable acheivable amount of regular play for one person, it still doesn't mean there's not a Right answer.
I recently posted this video which deals with primarily stardard deviations among some other issues discussed in this thread and as ER said the math does not lie,
. This is one of 5 videos in the series from the man that took down the house. I reccomend all 5 especially for the B&M BJ players. Hope it helps.

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#### MPB

##### Dormant account
oh come on, live a little!
If I felt like ignoring standard strategy I'd rather stand a 2 card 16 than draw to a t3 card one. IMHO having to draw with 16 is definitely one of he most frustating things in BJ.

#### happygobrokey

##### Dormant account
yeah i would want to surrender that guy! i think this is going to be a vp month for me... haven't played much at all in june, still waiting for all my funds to become available, stupid eft/cheques.