Blackjack probabilities for the players final score

Card distribution in the shoe
Card 2 3 4 5 6 7 8 9 X A Sum
Count:{$ sgtb.cards_total $}
Distribution:{$ sgtb.cards_dist.card_2 $}%{$ sgtb.cards_dist.card_3 $}%{$ sgtb.cards_dist.card_4 $}%{$ sgtb.cards_dist.card_5 $}%{$ sgtb.cards_dist.card_6 $}%{$ sgtb.cards_dist.card_7 $}%{$ sgtb.cards_dist.card_8 $}%{$ sgtb.cards_dist.card_9 $}%{$ sgtb.cards_dist.card_10 $}%{$ sgtb.cards_dist.card_1 $}%

 

 

Probabilities of the player last hand under the condition of the dealers first hand
DFH 2 3 4 5 6 7 8 9 X A
p(bust) {$ sgtb.player_bet.prob_bust.dfh_2 $} {$ sgtb.player_bet.prob_bust.dfh_3 $} {$ sgtb.player_bet.prob_bust.dfh_4 $} {$ sgtb.player_bet.prob_bust.dfh_5 $} {$ sgtb.player_bet.prob_bust.dfh_6 $} {$ sgtb.player_bet.prob_bust.dfh_7 $} {$ sgtb.player_bet.prob_bust.dfh_8 $} {$ sgtb.player_bet.prob_bust.dfh_9 $} {$ sgtb.player_bet.prob_bust.dfh_10 $} {$ sgtb.player_bet.prob_bust.dfh_1 $} {$ sgtb.player_bet.prob_bust.dfh_0 $}
p(BJ) {$ sgtb.player_bet.prob_BJ.dfh_2 $} {$ sgtb.player_bet.prob_BJ.dfh_3 $} {$ sgtb.player_bet.prob_BJ.dfh_4 $} {$ sgtb.player_bet.prob_BJ.dfh_5 $} {$ sgtb.player_bet.prob_BJ.dfh_6 $} {$ sgtb.player_bet.prob_BJ.dfh_7 $} {$ sgtb.player_bet.prob_BJ.dfh_8 $} {$ sgtb.player_bet.prob_BJ.dfh_9 $} {$ sgtb.player_bet.prob_BJ.dfh_10 $} {$ sgtb.player_bet.prob_BJ.dfh_1 $} {$ sgtb.player_bet.prob_BJ.dfh_0 $}
p(21) {$ sgtb.player_bet.prob_21.dfh_2 $} {$ sgtb.player_bet.prob_21.dfh_3 $} {$ sgtb.player_bet.prob_21.dfh_4 $} {$ sgtb.player_bet.prob_21.dfh_5 $} {$ sgtb.player_bet.prob_21.dfh_6 $} {$ sgtb.player_bet.prob_21.dfh_7 $} {$ sgtb.player_bet.prob_21.dfh_8 $} {$ sgtb.player_bet.prob_21.dfh_9 $} {$ sgtb.player_bet.prob_21.dfh_10 $} {$ sgtb.player_bet.prob_21.dfh_1 $} {$ sgtb.player_bet.prob_21.dfh_0 $}
p(20) {$ sgtb.player_bet.prob_20.dfh_2 $} {$ sgtb.player_bet.prob_20.dfh_3 $} {$ sgtb.player_bet.prob_20.dfh_4 $} {$ sgtb.player_bet.prob_20.dfh_5 $} {$ sgtb.player_bet.prob_20.dfh_6 $} {$ sgtb.player_bet.prob_20.dfh_7 $} {$ sgtb.player_bet.prob_20.dfh_8 $} {$ sgtb.player_bet.prob_20.dfh_9 $} {$ sgtb.player_bet.prob_20.dfh_10 $} {$ sgtb.player_bet.prob_20.dfh_1 $} {$ sgtb.player_bet.prob_20.dfh_0 $}
p(19) {$ sgtb.player_bet.prob_19.dfh_2 $} {$ sgtb.player_bet.prob_19.dfh_3 $} {$ sgtb.player_bet.prob_19.dfh_4 $} {$ sgtb.player_bet.prob_19.dfh_5 $} {$ sgtb.player_bet.prob_19.dfh_6 $} {$ sgtb.player_bet.prob_19.dfh_7 $} {$ sgtb.player_bet.prob_19.dfh_8 $} {$ sgtb.player_bet.prob_19.dfh_9 $} {$ sgtb.player_bet.prob_19.dfh_10 $} {$ sgtb.player_bet.prob_19.dfh_1 $} {$ sgtb.player_bet.prob_19.dfh_0 $}
p(18) {$ sgtb.player_bet.prob_18.dfh_2 $} {$ sgtb.player_bet.prob_18.dfh_3 $} {$ sgtb.player_bet.prob_18.dfh_4 $} {$ sgtb.player_bet.prob_18.dfh_5 $} {$ sgtb.player_bet.prob_18.dfh_6 $} {$ sgtb.player_bet.prob_18.dfh_7 $} {$ sgtb.player_bet.prob_18.dfh_8 $} {$ sgtb.player_bet.prob_18.dfh_9 $} {$ sgtb.player_bet.prob_18.dfh_10 $} {$ sgtb.player_bet.prob_18.dfh_1 $} {$ sgtb.player_bet.prob_18.dfh_0 $}
p(17) {$ sgtb.player_bet.prob_17.dfh_2 $} {$ sgtb.player_bet.prob_17.dfh_3 $} {$ sgtb.player_bet.prob_17.dfh_4 $} {$ sgtb.player_bet.prob_17.dfh_5 $} {$ sgtb.player_bet.prob_17.dfh_6 $} {$ sgtb.player_bet.prob_17.dfh_7 $} {$ sgtb.player_bet.prob_17.dfh_8 $} {$ sgtb.player_bet.prob_17.dfh_9 $} {$ sgtb.player_bet.prob_17.dfh_10 $} {$ sgtb.player_bet.prob_17.dfh_1 $} {$ sgtb.player_bet.prob_17.dfh_0 $}
p(16) {$ sgtb.player_bet.prob_16.dfh_2 $} {$ sgtb.player_bet.prob_16.dfh_3 $} {$ sgtb.player_bet.prob_16.dfh_4 $} {$ sgtb.player_bet.prob_16.dfh_5 $} {$ sgtb.player_bet.prob_16.dfh_6 $} {$ sgtb.player_bet.prob_16.dfh_7 $} {$ sgtb.player_bet.prob_16.dfh_8 $} {$ sgtb.player_bet.prob_16.dfh_9 $} {$ sgtb.player_bet.prob_16.dfh_10 $} {$ sgtb.player_bet.prob_16.dfh_1 $} {$ sgtb.player_bet.prob_16.dfh_0 $}
p(15) {$ sgtb.player_bet.prob_15.dfh_2 $} {$ sgtb.player_bet.prob_15.dfh_3 $} {$ sgtb.player_bet.prob_15.dfh_4 $} {$ sgtb.player_bet.prob_15.dfh_5 $} {$ sgtb.player_bet.prob_15.dfh_6 $} {$ sgtb.player_bet.prob_15.dfh_7 $} {$ sgtb.player_bet.prob_15.dfh_8 $} {$ sgtb.player_bet.prob_15.dfh_9 $} {$ sgtb.player_bet.prob_15.dfh_10 $} {$ sgtb.player_bet.prob_15.dfh_1 $} {$ sgtb.player_bet.prob_15.dfh_0 $}
p(14) {$ sgtb.player_bet.prob_14.dfh_2 $} {$ sgtb.player_bet.prob_14.dfh_3 $} {$ sgtb.player_bet.prob_14.dfh_4 $} {$ sgtb.player_bet.prob_14.dfh_5 $} {$ sgtb.player_bet.prob_14.dfh_6 $} {$ sgtb.player_bet.prob_14.dfh_7 $} {$ sgtb.player_bet.prob_14.dfh_8 $} {$ sgtb.player_bet.prob_14.dfh_9 $} {$ sgtb.player_bet.prob_14.dfh_10 $} {$ sgtb.player_bet.prob_14.dfh_1 $} {$ sgtb.player_bet.prob_14.dfh_0 $}
p(13) {$ sgtb.player_bet.prob_13.dfh_2 $} {$ sgtb.player_bet.prob_13.dfh_3 $} {$ sgtb.player_bet.prob_13.dfh_4 $} {$ sgtb.player_bet.prob_13.dfh_5 $} {$ sgtb.player_bet.prob_13.dfh_6 $} {$ sgtb.player_bet.prob_13.dfh_7 $} {$ sgtb.player_bet.prob_13.dfh_8 $} {$ sgtb.player_bet.prob_13.dfh_9 $} {$ sgtb.player_bet.prob_13.dfh_10 $} {$ sgtb.player_bet.prob_13.dfh_1 $} {$ sgtb.player_bet.prob_13.dfh_0 $}
p(12) {$ sgtb.player_bet.prob_12.dfh_2 $} {$ sgtb.player_bet.prob_12.dfh_3 $} {$ sgtb.player_bet.prob_12.dfh_4 $} {$ sgtb.player_bet.prob_12.dfh_5 $} {$ sgtb.player_bet.prob_12.dfh_6 $} {$ sgtb.player_bet.prob_12.dfh_7 $} {$ sgtb.player_bet.prob_12.dfh_8 $} {$ sgtb.player_bet.prob_12.dfh_9 $} {$ sgtb.player_bet.prob_12.dfh_10 $} {$ sgtb.player_bet.prob_12.dfh_1 $} {$ sgtb.player_bet.prob_12.dfh_0 $}
p(11) {$ sgtb.player_bet.prob_11.dfh_2 $} {$ sgtb.player_bet.prob_11.dfh_3 $} {$ sgtb.player_bet.prob_11.dfh_4 $} {$ sgtb.player_bet.prob_11.dfh_5 $} {$ sgtb.player_bet.prob_11.dfh_6 $} {$ sgtb.player_bet.prob_11.dfh_7 $} {$ sgtb.player_bet.prob_11.dfh_8 $} {$ sgtb.player_bet.prob_11.dfh_9 $} {$ sgtb.player_bet.prob_11.dfh_10 $} {$ sgtb.player_bet.prob_11.dfh_1 $} {$ sgtb.player_bet.prob_11.dfh_0 $}

This chart shows the probabilities of the players final score under the condition of the dealers first hand and with the implication that the player complies to the correct play strategy. This correct strategy is described in the charts for buy versus stay, for double versus buy and for splitting pairs.

This chart about probabilities is the fundamental tool to calculate the total expectations. If the dealers first hand is 2 through 6, chances to win are good for the player. On the other side, if the dealers first hand is 7 through Ace, chances to win are bad for the player. This means that if the low valued cards are gone, the player has an advantage. If the high values cards are gone, the dealer has an advantage. This is the principle for the hi-low counting system.

If you wonder, why the probability for a final score of 11 is slightly above zero, consider the situation where a player might double a score of 9 and he then receives a 2.