                             HOW PROBABILITY WORKS
                          The mechanics of Black Jack

If  you have played in the "Teach" mode you are no doubt curious about how the
advantages  and  probabilities are calculated. While  simple in its most basic
form,  probability  when  applied  to  a  game  such  as  Black  Jack  is very
computationally  intensive  and involves a  complete understanding of not only
the game, but the application of probability calculations to it.

Before  we discuss the game itself let's take  a moment to talk about a simple
probability  calculation that you can have fun  with. Suppose you want to know
the probability of drawing a two of any suit from a standard deck of 52 cards.
We  know  that  there  are four twos in  a  deck,  so  it's a simple matter to
calculate  your  chance  of  drawing  one. You  have  4  chances  in  52, or a
probability of .0769. Simple!

Now  lets go a step further with this example:  You have a five and a six, and
you  want  to know the probability of drawing  so  that you end up with a hand
value  of  twelve.  There's  only  one possibility  --  draw  an  ace.  So the
probability is 4/52, right? Wrong! Since you have already taken two cards from
the deck the answer is 4/50, or .080.

The problem becomes exponentially complicated when there are hundreds of paths
to  the desired outcome. Let's do another simple example: You have a three and
a  four,  and  you want to draw a hand  value  of  nine. There are two ways to
accomplish  this. You can simply draw a two  or you can draw two aces. Careful
here!  We  already know that the probability  of  drawing a two with two other
cards  already removed from the deck is .0800.

But you can also draw two aces. Calculate this probability like this:

                           P = (4/50)*(3/49) = .0049

Note  that  the  individual probabilties are  multiplied  together  to get the
overall  probability  of drawing two aces in  a  row. Also note that after the
first  one is drawn you have one less ace in the deck, and the deck has shrunk
to  49 cards for the second draw. You don't have a very good chance of drawing
two aces!

So  now we have two possibilities -- one  with a probability of .0800, and one
with  a probability of .0049. The overall  probability of drawing to a nine is
the sum of these two values: .0049+.0800 = .0849.

That's it!  You now know the fundamentals of calculating simple probabilities,
and this is the basis for Beat The House.

To  make  all  this  work  in  real  time  with  a  nearly  infinite number of
combinations  is  no  small task. The computer  must  keep  track of all cards
played,  how many are left in the  deck, and be able to calculate probabilties
for  all combinations that meet a set of rules. It must also be able to decide
if an ace should be counted as one or eleven.

DEALER PROBABILITIES AND STANDING
To  calculate  dealer probabilities is relatively  easy. In Beat The House the
dealer  must stand on any 17. Therefore  we process all combinations that lead
to  a  result  of  17 or greater.  The  dealer's  probabilties and the current
player's  score  are  all  that are  required  to  calculate  the advantage of
standing.

DOUBLING PROBABILITIES
It's  also easy to calculate probabilities  for doubling. Since the player can
only   draw  one  card  we  test  each  card  value,  2-11,  and  compute  the
probabilities for the hand values that result.

PLAYER PROBABILITIES
This  is  not so simple. Although the  basics  are the same, remember that the
player  can stand at any time. This problem is solved by first calculating the
"minimum standing hand", or the lowest value at which the player should stand.
Standing  values  are  tested from 12 to 19,  since  it would make no sense to
stand  on an eleven or try to draw  past 19. Once the optimum standing hand is
calculated  the computer calculates probabilities for all hands that meet this
new criterion using multiple draws. In practice Beat The House actually uses a
table  of values for the minimum standing values to improve speed. These might
not  be  completely  accurate  for all  situations,  but  the inaccuracies are
minimal.

                         CALCULATING THE BASIC STRATEGY

If  you have not already done so you should experiment with the Basic Strategy
Chart  generator available from the menu, and  then come back to this section.
The discussion below will probably make more sense.

The  Basic Strategy tells the player  where the highest advantage lies, either
standing, drawing, or doubling. This game does not allow splitting, so that is
not  attempted. The player's advantages for  each possibility are expressed as
multipliers  of  a  unit bet. In other words,  if  you bet one dollar, and the
advantage  for  standing is -.4357, you will  lose,  on the average, $0.44 for
each dollar bet with this hand if you stand.

Beat  The  House  uses  composition-dependent  basic  strategy,  as opposed to
total-dependent  strategy. Total-dependent strategy  requires a knowledge only
of   the   dealer's   upcard   and   the   total   of   the   player's   hand.
Composition-dependent strategy requires the values of all player cards as well
as  the  dealer's upcard. There is  little  difference in outcome for multiple
decks,  though  composition-dependent  strategy is  more  accurate  for single
decks.

Once  the  probabilities for standing, drawing,  and doubling are known we can
calculate  the  advantage for each. As a  simple  example of how this is done,
consider  the  probabilities for the dealer and  how they relate to the player
standing.  Remember  that  he  must draw until he  has  a  hand value of 17 or
greater or busts. Therefore we will have probabilities for hitting 17, 18, 19,
20, 21, and greater (bust).

Now  suppose that the sum of the probabilities  of the dealer getting 17 to 21
is  0.60, and the probability of busting is 0.40. If the dealer busts you win.
Therefore  if you have a hand that totals less than 17 you will win 40% of the
time  and  lose 60% of the time.  Your  advantage is then 0.40-0.60, or -0.20.
This  is the simplest example, though it  is only slightly more complicated if
you have a hand that totals 17 or greater.

The  advantages  of drawing or doubling are  done in a similar fashion, though
you  must account for the fact that you  will also bust and that the advantage
of doubling is in fact doubled, since you can win or lose twice your bet.

Your  strategy is calculated by comparing the advantages of standing, drawing,
or doubling and selecting the one that has the highest value.

That's the grand tour.  Now back to the game!





