Zero Margin Blackjack randomness test

September 14, 2012

I decided to review the Betfair zero margin blackjack game a few days ago; having heard bad reports about the game from time to time over the last few years, I thought it was time to gather some fresh statistics and see how they matched up to theoretical expectation.


I had originally planned to run ten thousand hands, but I ultimately gave up after just short of three thousand. I may pick up the test at a later date and struggle through to the original target. Of the total 2989 hands played, this is the breakdown of the numbers:

Wins	1030
Losses	1349
Ties	226
Double wins	124
Double losess	96
Double ties	19
Blackjacks	93
Suited blackjacks	40
5-card 21s	12

I have to admit at this point that my accounting skills were found a little wanting over the course of manually recording these hands - the overall tally from the above figures comes to -19.5, whereas my actual result was -25. So there is a 5.5 unit discrepancy between my records and the actual total. This shouldn't really have too much of a material impact. I think what may have caused the discrepancy is pressing the "redeal" button after a loss, looking down at my pad to record the result, with the dealer in the meantime taking a blackjack for another automatic loss which I then failed to recognise as a subsequent hand.


To start with, a 25-unit loss over 2989 hands represents a total -0.84% house edge. The standard deviation is 62.9, so my results lie comfortably within the boundaries of probability.


The percentages for the above figures are as follows:

Win	43.46%
Lose	48.34%
Tie	8.20%
Blackjacks	4.45%
Double win	56.36%
Double lose	43.64%

The theoretical percentages for win, lose and tie are:

Win: 43.3%
Lose: 47.8%
Tie: 8.9%

(See the Wong "streak" test table.)

My wins are a fraction above expectation, and the losses are a bit more again. The pushes are under. All the figures are close enough to the theoretical to not be of any concern.


Expectation for blackjacks in eight-deck is 4.74% compared to my overall 4.45%. This is a tad under, but if you look at the first table you'll see that 40 of the 124 total blackjacks were suited, which is a fraction over the theoretical. So, swings and roundabouts on that one. These figures all match up comfortably to expectation.

Of the doubles, you'd expect approximately 59% wins and 41% losses. My actual figures are a little outside this, but nothing to cause alarm.


After playing 2000 hands, I was worried on two counts: that the dealer was receiving more two-card 20s than me, and that I was busting with a ten card more than I should. In fact, on the latter I was absolutely certain that something was wrong. I decided therefore to record both these outcomes for the remainder of the test, which was a little short of 1000 more hands.


For the bust test, I recorded all cards received on totals greater than 12. As such, if I hit a 13 against 10 and received a 2, I recorded that card and then the next card also, as basic strategy requires a hit on the resulting 15. I recorded the four ten value cards together, without distinguishing between the ranks. The results are listed below:

Ace	2	3	4	5	6	7	8	9	10
22	27	12	27	28	18	29	26	31	111

There's a total of 331 bust cards, giving an average of 25.5 per card. The average for ace through 9, excluding the ten cards, is 24.5. For the tens, the average is 27.75. There are slightly more tens than the mean, but the same in fact goes for 2, 4, 5, 7, 8 and 9. Ace, 3 and 6 come in below average. As such, it does not appear from this test that my casual observation of excessive tens on bust hands was justified.

As to my concern about the two-card 20s: I recived 110, while the dealer received just 86, so again my observation doesn't seem justified.


Lastly, I looked at the pairs. With only 119 hands there really isn't enough data to draw any conclusions, but the results were quite bad:

Wins	33
Losses	50
Ties	6
Double wins	7
Double losses	10
Double ties	2
5-card 21s	1

If and when I pick up the test again, I'm going to look at doubles - towards the end of the run, I'd developed another hunch that I was constantly receiving small cards. It bears remembering that the two earlier hunchese I tested - two-card 20s and bust cards - did no show any anomalies upon examination, so I'd guess that the same would be true of a doubles test, though it remains to be verified.


I draw two conclusions from the above:

• The numbers do not suggest that the game is unfair.

• It's a very good idea to not invest too much certainty in "hunches"; I tested both mine, and they turned out to be unfounded.


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