Buzzo: BLACKJACK MATHEMATICS

[Buzzo]

Has anyone a good source, either a book or program, that will give the number of hands required on average with a 4 deck shoe, 6deck shoe, 8 deck shoe for a player to be dealt Blackjack. The number of hands played for a dealer and player both to have blackjack, the number of hands expected for 3 cards of the same value 2-A to be the first three cards dealt to a player, etc. ??????

[Don Schlesinger]

> Has anyone a good source, either a book or
> program, that will give the number of hands
> required on average with a 4 deck shoe,
> 6deck shoe, 8 deck shoe for a player to be
> dealt Blackjack. The number of hands played
> for a dealer and player both to have
> blackjack, the number of hands expected for
> 3 cards of the same value 2-A to be the
> first three cards dealt to a player, etc.
> ??????

These are pretty elementary problems in probability, so you might check out the probability section of any good bookstore or library.

As for the answers, they're fairly straightforward. Player probability for an untied natural is:

4-decks: 4.5437%
6-decks: 4.5323%
8-decks: 4.5266%

Respective probabilities that the dealer also has a natural are: 0.2129%, 0.2167%, and 0.2185%.

Add these numbers to the first group to determine the overall probability that a player has a natural. I'll do one of the three for you, as an example. In 6-deck, there are 6 x 52 = 312 cards.
So, to get a natural, you need, for example, and ace on the first card, which is 24/312 and then a ten-value on the second card, which is 96/311. Or, you might do it in the other order, so we have to multiply the product of our two fractions by 2.

So: 2 x (24/312) x (96/311) = 4608/97032 = 4.7489%.
Now, if you add the two numbers above, for 6-decks, you'll get the same answer, out to the last decimal, which may differ by one, for rounding. And, oh, by the way, the fraction we just got comes to one natural every 21 hands, so that should be fairly easy to remember! :-)

Three cards of the same value is somewhat trickier, because, for example, do splits count? If I start with 9-9, and split and get another 9 on the first card, does that count? Also, certain possibilities are eliminated, such as 10-10-10, unless you split tens also!

So, I'll pass on answering this one until you define things a little mnore clearly. Finally, if these questions are related to a certain game you're playing with various bonuses, why not just describe the game; chances are we've already done all the analysis.

Hope this helps.

Don