Rather than simulating blackjack, anyone can directly compute the expected values and overall strategy of blackjack directly using some applied mathematics. To build such analytical system, finding and solving some certain criteria is necessary, developing each algorithm to correctly solve for the optimal strategy.
What is prompting me to post this is to try to better understand a fundamental concept in blackjack : splitting. If it wasn't for this player option, blackjack would be much simpler to solve, but much boring to play (in my own normative opinion!) Therefore, developing a splitting algorithm is needed alongside solving the nitty-gritty maths associated with it.
I'll try my best to offer some maths to better express what it is I am trying to do; I will also be mixing descriptions of certain algorithms that would be ran on a computer in textual form.
1.) Computing the overall expected value for some strategy
The first thing I want to elucidate is how we can compute the overall strategy for blackjack. We can do this by way of computing E[X] as
E_opt[X|Y, j] = Σ[i = j; k] P[k|Y] * E_opt[Y|Y+k, k]; k < 11
That is, what is the conditional expectation of X given Y as the sum of the sum of each weighted conditional expectation for k cards drawn, all cards less than 11 points? Now, there is a problem. Assuming you evaluate this, we will go on for an infinite amount of time as we are not bounded by some restriction. Therefore, using the rules of blackjack, a hand cannot go above 21 points assuming that : a.) there are no aces that are 11 points that can be reduced to one (the soft hand rule), and b.) if there is an ace that is soft reduce it.
Taking these two rules we can compute the point total of hand Y given:
If a given hand is 11 or less and has an ace (a point of 1), add 10 points to the sum; otherwise, sum the points as is.
Now, if we were to enumerate the above E_opt equation and use the restriction above, how can one solve to find each unique subset plus the number of said subsets? We could simply have a computer do it, but is there a way to solve for these question? Is the above equation an accurate representation/description of an enumeration algorithm for finding/counting player hand and deck subsets?
2.) Finding an algorithm for splitting pair cards
The next step is to compute the expected value for splitting pair cards up to some k hands (usually up to 4, but we can do 3, or 2 as well.) The first thing we need to so is enumerate each subset for each split hand we compute. Say for example you are splitting a pair of 6's {66}. You split the hand to create two unique hands {6x, 6x}. You then enumerate the subset of player hands for each 6 and compute the overall expected value for splitting. You can do this up to 4 times : {6x, 6x, 6x} and {6x, 6x, 6x, 6x}.
The question remains is: how can we use the E_opt[X] equation above to find all subsets for each split, the number of subsets, as well as compute the optimal splitting expectation. Also take into account that sometimes, we may not get a split card as our first card as well as not being able to completely split up to our specified number of cards!
I apologise in advance if this is not clear enough for you. I sometimes have a difficult time expressing what I need in writing. I would also like to apologise for the lack of proper mathematical reasoning. My highest level maths I took was calc in my early college years.
TL;DR : How can I use mathematics to both find and count player/deck subsets for the game of blackjack as well as solving to find a splitting algorithm using maths to, again, find and count subsets as well as computing proper expected values.
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