Go for gold or stay with hope ?

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.

 

Numbers for when BR2 stands

The situation was originally as follows:

.........BkRll......Bet............Cards.......Action

P1.....1000......100...........10+6........Stand
You.....801......400...........10+3........Stand

Dealer................................6


Only 2 different things can happen:

(1) The dealer Busts a .4231 probability with a 6 up card and the situation for the last hand is:

You......1201......100.......(Best bet since you survive the lose-push swing)
P1........1100......500.......(P1 needs to win and might as well win big)

P1 wins 0.4329 of the time (using one player tables) so you win 1 - 0.4329 = .5671 of the time when the dealer busts.

(2) The dealer makes a hand a .5769 probability then the situation becomes:

You........401......401.......(Need to bet max and hope for a winning BJ)
P1..........900......100.......(Bets the minimum to survive any swing)

A BJ will occur .0473 of the time but not all BJs are winning since the dealer can get a BJ also. Winning BJ's occur .0451 of the time.

There as follow:

Dealer Busts...............0.4231 x 0.5671 = 0.2399
Dealer makes hand......0.5769 x 0.0451 = 0.0260

Total chance when standing on hand 9 = .2660

Of course this result assumes that P1 will play his hand according to the "basic strategy" for hitting and standing and no splitting of pairs. On hand 10 for example P1 should really stand with hard 15 vs a 10.


.....................................BlueLight

 

Thanks BlueLight for looking at this from a different angle. I'm always happy when my results agree with yours and my remarks here are intended only to possibly explain why our results differ and only because I would like some measure of comfort that my own results are not flawed.

My sim result was 0.4247. Possible influencing factors include:

• my use of an 8 deck simulated shoe with 75% penetration
• S17 vs H17 (I used S17)
• the occurrence of 2 card hand compositions other than 10+6 and 10+3 during the simulation
• insufficient simulator iterations (I used 100 million)

Here P1 does not necessarily win by simply winning his hand. There is a small chance that BR1 could split and double his way to victory which in turn is offset by the chance that P1 could prevent this by doubling/splitting. With both players playing optimal strategy, my number was a little lower at 0.5638.

I don't think that this situation is as simple as that. Even with his minimum bet, P1 could still fall victim to a full swing regardless of whether the dealer busts or not. I don't even know where to begin when it comes to breaking down the various possibilities.

P1's inability to lock out the full swing was the reason I had him betting enough to cover BR2's winning blackjack. A bet of 498 was a simple way to achieve this without giving up the 1/2 swing as well. My number for prevailing over P1 with this bet was substantially higher at 0.1381 (vs .0451).

Yes, playing strategy is a big factor. The players in my sims were playing optimal strategy, not because I think that anyone could actually do it perfectly, but because it probably more closely corresponds to an expert player's strategy than basic strategy.

 

Error in my assumptions

When the dealer makes a hand and tyhe bankrolls are 401 and 900 I overlooked that the minimum bet was 100.

Now I get the dealer busting with an 8 deck game with the cards removed as shown for the players and dealer upcard : Dlr Bust = .4383

Hero needs to bet only 100 on hand 10 to force P1 to win the hand.


When the dealer makes a hand then hero bets 401 and P1 should bet up to 498. The results are now as follows:

[(Dealer Bust) .4383] x [(P1 not win hand) .5686] = .2492
[(Dlr Mk Hand) .5617] x [(hero cover P1 BJ) .1233] =.0693

..........................................................Total.......0.3185

This result is for dealer hitting soft 17

For dealer standing on soft 17 I get Total = 0.3110


...............................BlueLight

 

Thanks! This is more in line with the result that I got (0.3189).