Blackjack statistics

[lucky21]

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]

Its not a direct answer to your question but you can calculate the standard deviation, see http://wizardofodds.com/blackjack/appendix4.html 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]

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]

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]

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:

http://wizardofodds.com/blackjack/appendix4.html

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]

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

[Random#rs]

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]

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 http://wizardofodds.com/casinos/casinobar.html , wizard of odds gives some other ideas of how to prove a blackjack game to be unfair.

[Random#rs]

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]

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.