As an example consider the choice of whether to draw or stand with (T,6) against a dealer 9. While relatively among the simplest borderline choices to analyze, we will see that precise resolution of the matter requires an extraordinary amount of arithmetic.
If we stand on our 16, we will win or lose solely on the basis of whether the dealer busts; there will be no tie. The dealer's exact chance of busting can be found by pursuing all of the 566 distinguishably different drawing sequences and weighting their paths according to their probability of occurrence.
Obviously a computer will be necessary to carry out the computations with satisfactory accuracy and speed. In Chapter Eleven there will be found just such a program.
Once the deed has been done we find the dealer's exact chance of busting is .2304, and it is time to determine the "mathematical expectation" associated with this standing strategy. Since we win .2304 bets for every .7696 ones we lose, our average return is .2304 - .7696 = -.5392, which has the interpretation that we "expect" to lose 54 cents on the dollar by hoping the dealer will break and not risking a bust ourselves.
This has been the easy part; analysis of what happens when we draw a card will be more than fivefold more time consuming. This is because, for each of the five distinguishably different cards we can draw without busting (A,2,3,4,5), the dealer's probabilities of making various totals, and not just of busting, must be determined separately.
For instance, if we draw a two we have 18 and presumably would stand with it. How much is this hand of T+6+2=18 worth, or in mathematician's language, what is our conditional expectation if we get a two when drawing? We must go back to our dealer probability routine and play out the dealer's hand again, only now from a 48 card residue (our deuce is unavailable to the dealer) rather than the 49 card remainder used previously. Once this has been done we're interested not just in the dealer's chance of busting, but also specifically in how often he comes up with 17,18,19,20, and 21. The result is found in the third line of the next table.
With our (T,6,2), or 18, we will win.1248 + .2324 = .3572, and lose .3553 + .1265 + .0565 = .5383. Hence our "conditional expectation" is .3572 - .5383 = -.1811. Some readers may be surprised that a total of 18 is overall a losing hand here. Note also that the dealer's chance of busting increased slightly, but not significantly, when he couldn't use "our" deuce.
Similarly we find all other conditional expectations.