double 14, too
I think that gonbog’s simulations are great for getting the right numbers for those who are curious to know the exact stats. Their even greater value is trying to apply them to see how they represent groups of situations and reveal “the inner” relationship of tournament plays. It builds a better tournament intuitions.
PlayHunter, you were correct that doubling on 14 is also a better play than standing. Having gronbog’s table for this case at your disposal you can see that standing on 13 advances P2 31.9% and doubling on 13 gives him 35.2%. To get the number representing win by doubling on 14 all you need to do is deduct from 35.2% (one-thirteen times dealer’s bust times %’s of game win with win/win outcome on the next to last hand.
35.2% – (7.7% x 26.2% x 85.1%) = 33.485%
33.485% advancement (double 14 on next-to-last hand) is better than standing on 14 – 31.89%
Here is detailed step by step how it works (it would help to have gronbog’s table open).
When you hit 13 once you will get 1/13th chance for each card, resulting in 3/13th stiff hands, 1/13th for each total 17 through 21, and 5/13th bust cards.
When you hit 14, in the same manner, you will get the same outcomes for 17 through 21 but one less stiff hand and one more bust.
There are five possible outcomes for your opponent’s and your hand: w/w, w/l, l/w, l/p, l/l.
When you lose your double you are eliminated, so the contributions from w/l and l/l are zero (in both cases) and you can omit it. You have to make a pat hand to have a chance for a push and since you getting the same amount of your pat hands whether you double 13 or 14 – l/p -these remain the same. Similarly, if your opponent loses then the dealer have a made hand and your pat hands (that can beat the dealer's pat hands) are the same for doubling both 13 and 14 –disregard w/l.
The only difference is when your opponent wins and you win (w/w); for this, since your opponent stayed on stiff hand, the dealer needs to bust. You will have the same amount of win with your hands made vs. dealer bust but since you have one (card) less of stiff hands (1/13th) of stiff hands that win with the dealer’s bust – that is the only difference between doubling 13 and 14.
Thus, we come with 1/13 x dealer bust x chances of game win when both players win round 9 of 10, which is 85.1%. The cost of one less stiff for the total of advancement is 1.715%.
This will apply to calculating doubling on higher totals 15 and 16.
You can see that doubling on 15 is 35.2% - (2 x 0.01715) = 31.8%
This is almost identical with 31.9% chance of winning the game by standing on any stiff.
S. Yama
PS
There are other methods of grouping the outcomes of the results for doubling 13 and 14 (depending on personal preferences of viewing it) but the conclusions are the same.
I think that gonbog’s simulations are great for getting the right numbers for those who are curious to know the exact stats. Their even greater value is trying to apply them to see how they represent groups of situations and reveal “the inner” relationship of tournament plays. It builds a better tournament intuitions.
PlayHunter, you were correct that doubling on 14 is also a better play than standing. Having gronbog’s table for this case at your disposal you can see that standing on 13 advances P2 31.9% and doubling on 13 gives him 35.2%. To get the number representing win by doubling on 14 all you need to do is deduct from 35.2% (one-thirteen times dealer’s bust times %’s of game win with win/win outcome on the next to last hand.
35.2% – (7.7% x 26.2% x 85.1%) = 33.485%
33.485% advancement (double 14 on next-to-last hand) is better than standing on 14 – 31.89%
Here is detailed step by step how it works (it would help to have gronbog’s table open).
When you hit 13 once you will get 1/13th chance for each card, resulting in 3/13th stiff hands, 1/13th for each total 17 through 21, and 5/13th bust cards.
When you hit 14, in the same manner, you will get the same outcomes for 17 through 21 but one less stiff hand and one more bust.
There are five possible outcomes for your opponent’s and your hand: w/w, w/l, l/w, l/p, l/l.
When you lose your double you are eliminated, so the contributions from w/l and l/l are zero (in both cases) and you can omit it. You have to make a pat hand to have a chance for a push and since you getting the same amount of your pat hands whether you double 13 or 14 – l/p -these remain the same. Similarly, if your opponent loses then the dealer have a made hand and your pat hands (that can beat the dealer's pat hands) are the same for doubling both 13 and 14 –disregard w/l.
The only difference is when your opponent wins and you win (w/w); for this, since your opponent stayed on stiff hand, the dealer needs to bust. You will have the same amount of win with your hands made vs. dealer bust but since you have one (card) less of stiff hands (1/13th) of stiff hands that win with the dealer’s bust – that is the only difference between doubling 13 and 14.
Thus, we come with 1/13 x dealer bust x chances of game win when both players win round 9 of 10, which is 85.1%. The cost of one less stiff for the total of advancement is 1.715%.
This will apply to calculating doubling on higher totals 15 and 16.
You can see that doubling on 15 is 35.2% - (2 x 0.01715) = 31.8%
This is almost identical with 31.9% chance of winning the game by standing on any stiff.
S. Yama
PS
There are other methods of grouping the outcomes of the results for doubling 13 and 14 (depending on personal preferences of viewing it) but the conclusions are the same.
