In How to Compute House Advantage, the odds for the outcome of the dealer’s hand when he must hit to 17, or for a player who follows the same rules in playing his hands, were given as:
The material that follows illustrates the calucaltions by which those odds were derived.
Odds for initial hands
The standard deck contains cards of thirteen possible values, so the odds of receiving a single card of any given value the same: there is a 1 in 13 chance (7.69%) a card will be of any value, and this is equal for every value. The net outcome for the value of the initial hand is as follows:
Granted, once the first card is dealt, the odds are minutely less that a card of identical value will be brawn as the second card in the hand, because there is one less of that value in the remaining deck. This difference would decrease the liklihood of drawing any even-numbered hand – but only by a minute percentage that becomes even more minute in multiple-deck games.
Odds of outcome for each hand
The odds of outcome, after all hits are taken, for each of these hands may likewise be calculated by applying the same likelihood (1 in 13 of each possiblevalue) to any hit that may added, up to the point that hits are no longer taken according to the rules by which a hand is played (in this case, upon reaching a total of 17).
As an example, an initial hand of fifteen (ten-five, nine-four, etc., omitting the four-ace combination that can fluctuate in value for the present time) would be hit once. Since the hand has not yet reached 17, there is a 100% chance a hit will be taken. For that hit, there is a 1 in 13 (7.69%) chance for each value:
This completes the calculations necessary to determine the total odds for a “hard” fifteen. The “soft” fifteen-an ace-four combination-is different because the original ace would revert to a value of one when the hand exceeds twenty-one. Instead of a 22 resulting in a busted hand, its value would revert to 12 and the player would continue to take hits, and it would require several more iterations to calculate the possible outcomes.
Odds of outcome for all hands
To determine the odds of outcome for the outcome of all hands, when the same rules are applied consistently, would be determined by calculating the possible outcome of each possible hand, multiplying it by the likelihood of drawing that hand, and adding them together.
This would require several pages of tables to illustrate completely, all derived from the basic tables shown above. To tabulate the odds for busting, it would be necessary to multiply the chances of drawing each hand by the chances of busting that hand, then add those results together for all hands.
For example, there is a 7.10% chance of drawing a hard 15 to begin, and a 58.58% chance you will bust before reaching any acceptable total (17+)-to arrive at a net chance of 4.1% that you will draw and bust a hard 15 from a freshly-shuffled deck.
That net chance would need to be added to the net chances of drawing and busting every other possible hand. These calculations have been done, and the results are …
Footnote A: Absolute Precision
In deference to those who rejoice at any opportunity to niggle, it would require some minor adjustments to compute these odds with absolute precision. For example, it is less likely to draw a paired hand than any other because, once the first card has been removed, there is one fewer card of the exact same value left in a single deck when compared to cards of every other value. It is also less likely that the next hit on a six-nine hand (15) would be a six or a nine, so the calculations would have to be done for every unique combination rather than every unique value to arrive at absuolutely precise odds.
However, these differences are minute. The odds of drawing a paired hand is 0.591% if only the values of the cards are considered, which would be 1.8% less likely to occur (0.580%) in a single-deck game, 1.2% less likely (0.589%) in a double-deck game, and so on, with the difference becoming more minute as decks are added. The net result is a difference of a few thousandths of a percent per pair.
While it would, indeed, derive a more precise estimation of odds and strategy to adjust the calculation for cards that have already been dealt, te result would affect only 1 in 100,000 hands in most games. A difference this minute does not lead to any significant differences, hence it is hardly worth considering in practical application of the results.