In Wong's book "Casino Tournament Strategy" on page 131 he gives player outcome probabilities for 1,2,3,4 players. The tables are given to only 2 decimal places, probably because the tables are the result of simulated runs of thousands of hands. Below I give the exact probabilities for one player for an infinite deck. The player makes the correct hit or stand decisions against the various dealer up cards. No splitting or doubling is considered. The result was done by examining every possible player starting 2 cards and dealer upcard. Extra care by the computer program was needed when the dealer had a 10 or A for an upcard. For example when we decide to stand with 10+9 against a dealer A. The outcome for this situation is W=.4906 P=.2069 L=.3026. However this would omit the times the dealer has a BJ. The outcome for the above situation before the dealer checks for BJ would then be W=.3396 P=.1432 L=.5172. Also note that the probability for a player winning BJ is .0451 compared for a player BJ of .0473. (Not all player BJ's are winning since the dealer may have a BJ as well) The tables below show a Win whether it's a BJ bonus win or single win. Then the table at the right separates the 2-1 BJ Win and a single win. I would welcome any confirmation or differences other people may have. Infinite Deck Dealer Hits Soft 17 .........................................Win(2)....... .0451 Win...... .4320 ....................Win(1)....... .3869 Push...... .0870 ....................Push......... .0870 Lose...... .4810 ....................Lose......... .4810 Infinite Deck Dealer Stands on Soft 17 .........................................Win(2)...... .0451 Win...... .4329 ....................Win(1)...... .3878 Push...... .0874 ....................Push........ .0874 Lose...... .4797 ....................Lose........ .4797 ...........................BlueLight
good job BlueLight, thank you for interesting information. Actually I have slightly different numbers (roundings 'til decimals): P(win) = 43.4% P(push) = 8.6% P(lose) = 48.0% Same conditions applied: H17; No DD; No Split; No Insurance, No Surrender. These probabilities include Dealer's possible BlackJacks. As you've mentioned in your post for Dealer's T and A. we should take into consideration also possible BJ. Without any computer programming we can transfer the numbers we have for infinitive deck (combinatorial) - "NO BJ" into the case when BJ is possible. Here we have: ..............17........18.........19..........20........21.......bust A ..........8.30.....20.68.....20.68.....20.68.....9.51.....20.15 (No BJ) A...........5.75.....14.32.....14.32.....14.32....37.34....13.95 (BJ is possible) All we did is just multiply first line by 9/13 and then added 4/13 (probability of BJ) to the changed probability of 21 (9.51 x 9/13). Same logic applies for Dealer's T-up. ..............17........18..........19.........20..........21......bust T...........12.07...12.07.....12.07......37.07......3.74....22.98 (No BJ) T...........11.14...11.14.....11.14......34.22.....11.14...21.22 (BJ is possible) We multiplied first line by 12/13 and then added 1/13 (probability of BJ) to the changed probability of 21 (3.74 x 12/13). By the way it'll be interesting to check our numbers of making hand by the Dealer and/or bust (H17) D(17) = 13.33 D(18) = 14.15 D(19) = 13.55 D(20) = 18.22 D(21) = 7.47 D(BJ) = 4.73 D(bust) = 28.55 Total 100%
Confirmation! I'm glad someone has come up with figures similar to mine. It means that my results are very likely right. The only confirmation I had was when the program showed a winning player BJ at .0451. The next step is to make a table for 2 players. Then I can figure bets more accurately with the BJ bonus figured in. The 2 player table would work well for "all in" type of tournaments since the player doesn't have to double down or split to get more money into the bet circle. He simply bets as much of his bank roll as needed. (This is not always true for "all in" tournaments; sometimes with a chip lead we need to double down to correlate our bets with an opponent or when trailing we double down to uncorrelate our bets with an opponent) By working backward the next to last hand bets can be figured out and so on for the 3rd to last hand etc. .........................BlueLight
Dealer probabilities for various up cards. Infinite Deck.......Dealer H17S18........Up = None Yet ............................................B.Peek..........A.Peek ......BlkJck..............22........... .04734........... ------ ............................21........... .07474.......... .07846 ............................20........... .18226.......... .19131 ............................19........... .13547.......... .14221 ............................18........... .14151.......... .14854 ............................17........... .13326.......... .13988 .....Plyr stiff............16............ ------........... ------ .....Delr bust...........15........... .28542........... .29961 .....Plyr bust...........14............ ------............ ----- Infinite Deck.......Dealer H17S18........Up = A ............................................B.Peek.............A.Peek ......BlkJck.............22............ .30769............. ------ ...........................21............ .06628............ .09574 ...........................20............ .14320............ .20685 ...........................19............ .14320............ .20685 ...........................18............ .14320............ .20685 ...........................17............ .05749............ .08305 .....Plyr stiff...........16............. ------............ ------ .....Delr bust..........15............ .13892............. .20066 .....Plyr bust..........14............. ------..............------ Infinite Deck.......Dealer H17S18.......Up =10 ............................................B.Peek.............A.Peek .......BlkJck...........22.............. .07692............. ------ ..........................21.............. .03450............ .03738 ..........................20.............. .34219............ .37071 ..........................19.............. .11142............ .12071 ..........................18.............. .11142............ .12071 ..........................17.............. .11142............ .12071 .....Plyr stiff..........16............... ------............ ------ .....Delr bust.........15.............. .21211............. .22978 .....Plyr bust.........14............... ------............. ------ ........................BlueLight
