Calculating BJ Basic Strategy Probabilities

Hello, everyone! I just joined the Wizard of Vegas, and quickly browsed the "BJ" and "Math" forums, but did not see this discussed, so here goes...

I understand the BJ Basic Strategy, but I would like to see the probabilities behind each stratetgy, and (hopefully) a reasonable calculation method for these probabilities. For example, if I have a 14-count, and the dealer is showing a '5', the basic strategy says to 'stay' (and I do this ALL the time when I play). I understand this to mean that I have a higher probability winning the hand by staying (hoping that the dealer will bust), than by hitting. But what is the probability of the dealer busting, when showing a '5', and how is this calculated? Versus, what is the probability of me improving my hand by hitting (this is actually pretty easy to calculate)? And this is an easy case. Or have these probabilities really just been estimated by Monte Carlo type simulations?

By way of background, I have a Masters degree in Mathematics (OK, it's 30 years old, but math doesn't change), with a focus on P's and C's (permutations and combinations), which are obviously related to probability and game theory. I am also adept at using Excel and computer programming. So I am interested in any sort of formulae or algorithms that could help me out here.

Thanks in advance!

A copy of the Wizard's infinite deck spreadsheet can be found in google docs. Some good threads are here, here, and here.

Very cool - I especially enjoy the spreadsheet. THANKS!

It's probably the same thing that's in the spreadsheet, but the first place I go is to the Wizard's Appendix 1: here.

For the condition you specifiy, the expected return for standing is -0.1672, and the expected return for hitting is -0.3214. So it is clear that the correct play is to stand, and the difference is significant. Also consider the case of a 16 against a face card. Standing yields -0.5405, while hitting yields -0.5398. So the correct play is to hit, but you can see that this decision really doesn't make that much of a difference.

Quote: pjlatimer

Very cool - I especially enjoy the spreadsheet. THANKS!

Try to do this simple problem on a spreadsheet.

From the Wizard's spreadsheet for an infinite deck extract the computation that comes up with the calculation dealer 7, player 17,if you stand you should have an expected value of -0.1068. Although that is a negative E.V., the alternatives are worse (hitting, doubling, or surrender=-0.500).

Now assume a single deck, and try to get the following E.V. if the player stands:
dealer 7, player has a 9 & 8 for a sum of 17. E.V.=-0.122900
dealer 7, player has a 10 & 7 for a sum of 17. E.V.=-0.121287

In this particular case it won't matter if the rule for the dealer is "stand on soft 17" or "hit on soft 17"

You will be surprised at how many rows of calculations you need for this relatively simple case. The mathematics is tedious, but it isn't brain surgery. But just start at this one case, and you will see how the calculations are derived.

Counting theory is a matter of getting every single possibility, and counting the ones you win versus the ones you lose.

Just say what is the probability of the dealer down card being a 10. If this happens then the hand is a push
For an infinite deck the odds are 4/13 .
For a single deck the odds are 16/49 for the first case since there is a 7, 8 and 9 out of the deck.
For a single deck the odds are 15/50 for the 2nd case since there is a 7, 7 and 10 out of the deck.

Even if you are going to program the solution, you should work out at least one case by hand, and it will give you a check to test out your algorithm,

Quote: pjlatimer

I understand the BJ Basic Strategy, but I would like to see the probabilities behind each stratetgy, and (hopefully) a reasonable calculation method for these probabilities. For example, if I have a 14-count, and the dealer is showing a '5', the basic strategy says to 'stay' (and I do this ALL the time when I play). I understand this to mean that I have a higher probability winning the hand by staying (hoping that the dealer will bust), than by hitting. But what is the probability of the dealer busting, when showing a '5', and how is this calculated? Versus, what is the probability of me improving my hand by hitting (this is actually pretty easy to calculate)? And this is an easy case. Or have these probabilities really just been estimated by Monte Carlo type simulations?

The way I did it (BTW, I have a thirty-year-old degree of my own, in Computer Science, and have a strong math background) was:

First, for each combination of number of decks and whether the dealer hits or stands on soft 17, I went through every possible deal of cards from a full deck for the dealer's hand to determine the probabilities of getting 17, 18, 19, 20, 21, or bust for each up card value.

Then, to calculate the player's strategy, for each of the ten upcard values, I started with "hit or stand on hard numbers", starting with 21. If you stand on 21, your expected win = the probability of the dealer getting 17 through 20 or busting; if you hit 21, your expected win = 1/13 x (your expected win with 22) + 1/13 x (your expected win with 23) + ... + 4/13 x (your expected win with 30). Of course, all of those "expected win" values are -1. If the "expected win by staying" > the "expected win by hitting", you stay; otherwise, you hit.
Do the same thing with 20, then 19, then 18, and so on down to 12. (You work from 21 down because you have to know the "expected win" values for all values that are higher than the one you are working on.)
You then work on soft 21 through soft 12, then 11 through 4.

Note that there is a slight fudge - if the dealer's up card is, say, 5, then you shouldn't multiply the value you get if you draw a 5 by 1/13 as there aren't necessarily as many 5s in the deck as there are other cards.

Calculations for standing assuming infinite deck

These are the same calculations that the Wizard does in his spreadsheet in Appendix 1 of the WOO Blackjack section. I tried very hard to simplify the equations used so that they are relatively uniform. The same numbers are in the Wiz's spreadsheet, but he didn't try as hard to use fewer formula, and he has some bookkeeping to invert matrices and other things. I am hopeful you will find mine a little easier to follow.

But I only started the problem, I didn't finish it.