Blackjack Mechanic Probabilities

[d3nial]

Trying to figure the probabilities of a modified blackjack mechanic:

Am I right in assuming that the average value of a single card drawn from a 52 card deck is: (2+3+4+5+6+7+8+9+(4*10)+6)*4/13*4

(4*10) are the 4 x (10 + picture cards)
6 (at the end) represents an Ace being either 1 or 11 [(1+11)/2 = 6?]

This formula would give a value of: ~6.92 (please correct me if I'm wrong).

I have no idea how to calculate the average value of subsequent cards - is it just the same?

If my calcs are correct, one card will be worth ~6.92, 2 cards will be worth ~13.84 and 3 cards will be worth ~20.76?

Any help much appreciated. Thanks.

Daniel

[NN]

What exactly are you after?

Aces high, mean is 7.31
Aces low, mean is 6.54
Aces randomly high or low, mean is 6.92

but if youre trying to draw cards to a target number, like in blackjack, then means arent that relevant.

[M. J. Young]

The problem with card-based probabilities is that the probabilities have to be recalculated after every draw.

The simple illustration is that the odds of drawing the Ace of Spades from a fifty-two card deck (no jokers) is 1/52. The odds of drawing any ace from that deck is 4/52, or 1/13, about 7.7%. The problem is that having drawn one ace, the probability of drawing a second ace from the same deck is 3/51, about 5.9%; the odds of drawing a specific non-ace value card as the second card are 4/51, 7.8%. On the third draw, the denominator goes to 52, and the odds on the numerator depend on how many of that card have been drawn already.

I don't know if this helps, but these are the odds of drawing a specific number, both for individual values and for "face cards including ten" (of which there are sixteen in the deck rather than four), based on the number of cards drawn from the deck already (across the top) and the number of those which are of that type (down the side) (rounded to two decimals as percentages):

Code Select Expand

   Number Drawn:  0     1     2     3     4     5     6     7     8
Type:
Any Number         7.69  7.84  8.00  8.16  8.33  8.51  8.70  8.89  9.09
Face Card         30.77 31.37 32.00 32.65 33.33 34.04 34.78 35.56 36.36
Any Number, 1 drawn      5.88  6.00  6.12  6.25  6.38  6.52  6.67  6.82
Face Card, 1 drawn      29.41 30.00 30.61 31.25 31.91 32.61 33.33 34.09
Any Number, 2 drawn            4.00  4.08  4.17  4.26  4.35  4.44  4.55
Face Card, 2 drawn            28.00 28.57 29.17 29.79 30.43 31.11 31.82
Any Number, 3 drawn                  2.04  2.08  2.13  2.17  2.22  2.27
Face Card, 3 drawn                  26.53 27.08 27.66 28.26 28.89 29.55
Face Card, 4 drawn                        25.00 25.53 26.09 26.67 27.27
Face Card, 5 drawn                              23.40 23.91 24.44 25.00
Face Card, 6 drawn                                    21.74 22.22 22.73
Face Card, 7 drawn                                          20.00 20.45
Face Card, 8 drawn                                                18.18

This should give you a start. The value is the number of that type of card remaining in the deck over the total number of cards remaining in the deck (times one hundred to convert to percentages).

--M.  J. Young
Edited to reduce width of table by removing columns for ninth and tenth draws.