New and improved Table 4.3 ready

After trying to find another way to speed up a 2 player program a method was found. It took just 24 seconds to generate a 2 player chart other than 2 hours. Using the same methology a 3 player chart was generated in 26 seconds rather than a projected 720 hours using the old method. The results are given below:

Dealer.....HS=17.....SS=18

....Result.....................Probability............Permutations

Win...Win...Win............. .22763....................1
Win...Win...Psh............. .00927....................3
Win...Win...Lse............. .05191....................3
Win...Psh...Psh............. .00320....................3
Win...Psh...Lse............. .01143....................6
Win...Lse...Lse............. .05595....................3
Psh...Psh...Psh............. .00178....................1
Psh...Psh...Lse............. .00702....................3
Psh...Lse...Lse............. .03266....................3
Lse...Lse...Lse............. .22198....................1



Dealer.....HS=17.....SS=17

....Result......................Probability..........Permutations

Win...Win...Win.............. .22634..................1
Win...Win...Psh.............. .00956..................3
Win...Win...Lse.............. .05244..................3
Win...Psh...Psh.............. .00326..................3
Win...Psh...Lse.............. .01156..................6
Win...Lse...Lse.............. .05619..................3
Psh...Psh...Psh.............. .00178..................1
Psh...Psh...Lse.............. .00700..................3
Psh...Lse...Lse.............. .03246..................3
Lse...Lse...Lse.............. .21979..................1


Note that the permutation column gives the total number of sequences a result can have. For instance a Psh...Win...Psh result is not given in the column; however it would be the same as the Win...Psh...Psh result. The same applies to a Psh...Psh...Win.

As a check these figures can be used to find a 2 player table as follows:

Win...Win...Win..................... .22634
Win...Win...Psh..................... .00956
Win...Win...Lse..................... .05244

Sum for a 2 player Win...Win... .28834

This agrees with the .2883 figure given in the 2 player tables previously.

The figures closely agree with Wong's table 4.

.......................From Table.................Wong
All win................. .23634 ................... .23


2 Win other loses or pushes

.......................From Table.........................................Wong
WWP.................. .00956........x3........= .02868
WWL.................. .05244........x3........= .15732

...........................................Sum....... .18600 ............ .19

The tables here give separate results for WWP and WWL while Wong does not, probably because he keeps the table simple to 2 decimal places.



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

 

Thanks for that

Does this speed boost mean that you can now go on to surpass Wong and produce tables 4.5, 4.6 etc.?

I've always found the issue of permutations confusing when looking at Wong's table, partly because he talks of player A and player B's results in the two-player case, but then switches to the just deal in the total numbers of players thereafter, and partly because ..... well, I'm easily confused.

I find it easier to understand the way you have presented it. So, for instance, the probability that two players win, while the other pushes or loses is

Code:

WWP = .00956 x3
WWL = .05244 x3  
-----------------
      .186   
-----------------
(agreeing with Wong's .19)

Whereas, the probability that I win, while one other wins and the third pushes or loses is -

Code:

WWP = .00956 x2
WWL = .05244 x2  
-----------------
      .124   
-----------------
(because two of the three permuations in each case have me winning my hand)

Apologies, if I'm stating the obvious!

 

You got that correct

To clarify your situation of you winning and a 2nd player wins and a 3rd player pushes or loses is:

...........You.........P2..........P3...........Probability

............W..........W............P............. .00956
............W..........P.............W............ .00956
............W..........W............L............. .05244
............W..........L.............W............ .05244

...........................................Total.... .12400


The key here is there are fewer permutations when you specify that "YOU" win and another player wins and a 3rd pushes or loses. There must be a W under the "You" column.

For any situation wher 2 players win and a 3rd pushes or loses is as follows:

Situation....................Table look up

W...W...P...........=........W...W...P
W...P...W...........=........W...W...P
W...W...L...........=........W...W...L
W...L...W...........=........W...W...L
P...W...W..........=.........W...W...P
L...W...W..........=.........W...W...L


Tables 4.4, 4.5, and 4.6 should be produseable.

I don't know how Wong got his tables so I think a random number generator to check these tables needs to be done. For a single player against the dealer (H17S18) I got after 1 million deals is:

Win........ 43.17%
Push....... 8.82%
Lose....... 48.01%


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

 

Table 4.3 with BJ

Below is table for 3 player outcome with blackjacks wins (no dealer BJ) listed as a 4th possibility and separate from from ordinary single wins. In the earlier table a W could be a win or BJ win. Dealer H17S17 (stand on hard 17 and soft 17). Here J w w = .01181 is the probability that P1 receives a winning BJ (no dealer BJ) and P2 and P3 win single bets. The probability that any player will get a BJ and the other 2 will win single bets is 3 x .01181 = .03543.

....P1....P2...P3..............Prob.............Permutations

.....J.....J.....J............. .00010............... 1
.....J.....J....w............. .00087............... 3
.....J.....J....P.............. .00019............... 3
.....J.....J....L.............. .00097.............. 3
.....J....w....w............. .01181.............. 3
.....J....w....P.............. .00096.............. 6
.....J....w....L.............. .00472.............. 6
.....J....P....P.............. .00056.............. 3
.....J....P....L.............. .00232.............. 6
.....J....L....L.............. .01257.............. 3
....w....w...w............. .18820............... 1
....w....w...P.............. .00744............... 3
....w....w...L.............. .04203............... 3
....w....P....P............. .00269............... 3
....w....P....L............. .00925............... 6
....w....L....L............. .04362............... 3
....P....P....P............. .00178............... 1
....P....P....L............. .00700............... 3
....P....L....L............. .03246............... 3
....L....L....L............. .21979................ 1

..........................................Total..... 64


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