Probabilities in the first four cards in blackjack (Appendix 13)

Last update: June 24, 2001

In playing blackjack online one problem I often face is not knowing how many decks are being used. This is a particular problem with Real Time Gaming casinos. The help files often do not indicate this rule, as well as other rules, and customer support are notorious for giving incorrect information on their own rules. So I devised a test to help determine the number of decks. This test is based on the player's first two cards and the dealer's first two.

The following table shows the probability for various configurations of the initial four cards in blackjack. Note the probabilites for a suited pair. Of all the hands I feel this is the best to test for, given both it's frequency and correlation to number of decks.

4-Card Hand	1 deck	2 decks	4 decks	6 decks	8 decks
4 singletons	0.676110	0.63692	0.618504	0.612530	0.609573
Non-suited pair	0.304250	0.286614	0.278327	0.275638	0.274308
Suited pair		0.047769	0.069582	0.076566	0.080006
Two non-suited pairs	0.010372	0.009771	0.009488	0.009397	0.009351
Two suited pairs		0.000271	0.000593	0.000725	0.000796
Two pair - 1 suited		0.003257	0.004744	0.00522	0.005455
3 of a kind - 3 suits	0.009220	0.008685	0.008434	0.008353	0.008312
3 of a kind - 2 suits		0.006514	0.009488	0.010441	0.010910
3 of a kind - 1 suit			0.000527	0.000773	0.000909
4 of a kind - 4 suits	0.000048	0.000045	0.000044	0.000044	0.000043
4 of a kind - 2 suits (3&1)			0.000033	0.000048	0.000057
4 of a kind - 2 suits (2&2)		0.000017	0.000037	0.000045	0.000050
4 of a kind - 3 suits		0.000136	0.000198	0.000218	0.000227
4 of a kind - 1 suit			0.000001	0.000002	0.000003
Total	1	1	1	1	1

To determine the number of decks in an online blackjack game keep a tally of both the total number of hands played and the number of suited pairs. Only count a hands as a suited pair if the other two are singletons. For example one suited pair and one non-suited pair does not count. In a single deck game the ratio of suited pairs to total hands will obviously be zero. In double deck this ratio will be about 4.8%. In a 4-deck game the ratio increases to 7.0%. After that the differences are too subtle are to tell without a gigantic sample.

Of course if you ever notice three of the same card on the screen at once that rules out a double deck game immediately. Despite my lack of faith in customer support knowing their own rules I would suggest at least asking. If they give you an incorrect answer, and you can prove it, you may get some free money in your account as a way of thanks. This has happened to me several times.

Unfortunately it takes a fairly large sample size to have confidence in the number of decks between 2 and 4. After 250 hands the probability that the sample mean in a 2-deck game will be greater than 6.96% (the 4-deck theoretical mean) is 5.29%. Likewise the probability that the sample mean in a 4-deck game will be less than 4.78% (the 2-deck theoretical mean) is 8.76%. Increasing the sample size to 500 these numbers become 1.11% and 2.76%. At 1000 the numbers are 0.06% and 0.34%.

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