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Last Updated: August 22, 2016

Probability of a Win at Various Mid-game Stages

Introduction

This appendix shows the probability of a banker, player, and tie win at various mid-game stages. All probabilities are based on eight decks.

After Two Cards

In big table baccarat there is a break between the player's first two cards being exposed and the banker's first two cards. The following table displays the probability of each possible final outcome based on the player's two-card total, and on an 8-deck game.

Probabilities after Two Player Cards

Player
Total
Player Win Banker Win Tie Win
0 0.278792 0.654627 0.066581
1 0.282833 0.647252 0.069915
2 0.290303 0.634207 0.07549
3 0.303386 0.615539 0.081075
4 0.322084 0.579983 0.097933
5 0.358488 0.537849 0.103663
6 0.430362 0.427801 0.141838
7 0.572657 0.285046 0.142297
8 0.810911 0.094938 0.094151
9 0.905475 0 0.094525
Total 0.446247 0.458597 0.095156

After Three Cards

The following three tables display the probability of each possible final outcome given the totals after the three four cards are exposed (two player cards and one banker card), based on an 8-deck game.

Probabilities of Player Win after Three Cards — 8 Decks

Banker
Card
Player Total
0 1 2 3 4 5 6 7 8 9
0 0.306893 0.322192 0.33142 0.345345 0.36393 0.397013 0.478841 0.609314 0.846542 0.923724
1 0.303056 0.307637 0.327876 0.341492 0.359810 0.392935 0.479214 0.609687 0.846552 0.922813
2 0.29888 0.303544 0.313111 0.337875 0.356007 0.388382 0.4794 0.60969 0.845637 0.922813
3 0.297826 0.301239 0.309319 0.321807 0.349428 0.381111 0.481816 0.608594 0.845637 0.922813
4 0.293968 0.289011 0.288859 0.293017 0.315714 0.404964 0.481161 0.611379 0.84804 0.922813
5 0.281471 0.273613 0.270579 0.285812 0.305842 0.341949 0.478745 0.60915 0.845627 0.922813
6 0.264707 0.254029 0.262176 0.274764 0.293293 0.325144 0.318854 0.662697 0.845637 0.922813
7 0.242915 0.242836 0.248376 0.260008 0.274153 0.302702 0.318691 0.43136 0.845637 0.922813
8 0.207034 0.207542 0.213991 0.223921 0.238433 0.267405 0.320167 0.430432 0.614096 0.922813
9 0.207713 0.208903 0.214243 0.224154 0.23862 0.267514 0.319980 0.432631 0.616509 0.691264

Probabilities of Banker Win after Three Cards — 8 Decks

Banker
Card
Player Total
0 1 2 3 4 5 6 7 8 9
0 0.61261 0.603259 0.589483 0.570967 0.538812 0.501127 0.391797 0.261131 0.077186 0
1 0.62701 0.606708 0.593152 0.574934 0.542666 0.504937 0.390304 0.259829 0.076262 0
2 0.631112 0.621728 0.596677 0.578569 0.547237 0.509478 0.390292 0.259827 0.077177 0
3 0.633433 0.625444 0.612958 0.585173 0.554372 0.504024 0.390476 0.261176 0.077177 0
4 0.645421 0.645692 0.641575 0.618955 0.530124 0.488289 0.388611 0.258021 0.077177 0
5 0.660972 0.664253 0.64884 0.628854 0.594971 0.502523 0.390839 0.260619 0.077186 0
6 0.680465 0.672742 0.659888 0.642874 0.610422 0.574009 0.337292 0.224806 0.077177 0
7 0.691696 0.686316 0.676197 0.660356 0.632850 0.599213 0.568625 0.224781 0.077177 0
8 0.744516 0.739623 0.728726 0.71421 0.685859 0.65247 0.569553 0.456697 0.077173 0
9 0.743823 0.738262 0.728303 0.713851 0.685438 0.652171 0.567151 0.454615 0.308707 0

Probabilities of Tie Win after Three Cards — 8 Decks

Banker
Card
Player Total
0 1 2 3 4 5 6 7 8 9
0 0.080497 0.07455 0.079097 0.083687 0.097258 0.101861 0.129363 0.129555 0.076271 0.076276
1 0.069935 0.085655 0.078972 0.083574 0.097524 0.102128 0.130482 0.130484 0.077186 0.077187
2 0.070008 0.074729 0.090212 0.083556 0.096756 0.10214 0.130308 0.130484 0.077186 0.077187
3 0.068741 0.073317 0.077723 0.093020 0.096200 0.114865 0.127708 0.130231 0.077186 0.077187
4 0.060611 0.065297 0.069566 0.088028 0.154162 0.106747 0.130229 0.1306 0.074784 0.077187
5 0.057557 0.062134 0.080581 0.085333 0.099188 0.155528 0.130415 0.130231 0.077186 0.077187
6 0.054828 0.073229 0.077936 0.082361 0.096284 0.100847 0.343854 0.112498 0.077186 0.077187
7 0.06539 0.070848 0.075427 0.079636 0.092996 0.098085 0.112684 0.343859 0.077186 0.077187
8 0.04845 0.052835 0.057282 0.061868 0.075708 0.080125 0.11028 0.112871 0.308731 0.077187
9 0.048464 0.052835 0.057454 0.061995 0.075942 0.080315 0.112869 0.112755 0.074784 0.308736

After Four Cards

The following three tables display the probability of each possible final outcome given the totals after the first four cards are dealt, based on an 8-deck game.

Probability of Banker Win

Banker
Total
Player Total
0 1 2 3 4 5 6 7
0 0.427793 0.40973 0.391882 0.374037 0.356195 0.33765 0.231516 0.15434
1 0.495459 0.426929 0.409234 0.391336 0.373435 0.354619 0.230281 0.153216
2 0.513308 0.496004 0.42693 0.409025 0.391346 0.37253 0.23028 0.153224
3 0.525182 0.513697 0.501957 0.438649 0.4267 0.354636 0.230271 0.154128
4 0.578664 0.60282 0.627004 0.591578 0.326054 0.289715 0.230277 0.153211
5 0.638716 0.674654 0.650906 0.62815 0.604411 0.348888 0.231193 0.154125
6 0.721859 0.710369 0.698539 0.687609 0.675001 0.662331 0 0
7 0.769393 0.769718 0.769708 0.768806 0.769725 0.76881 1 0

Probability of Player Win

Banker
Total
Player Total
0 1 2 3 4 5 6 7
0 0.427793 0.495459 0.513308 0.531152 0.548995 0.567741 0.692221 0.769393
1 0.40973 0.426929 0.496004 0.513697 0.531594 0.550413 0.69265 0.769718
2 0.391882 0.409234 0.42693 0.496004 0.513892 0.532501 0.692656 0.769708
3 0.385978 0.397289 0.409025 0.420977 0.484285 0.497143 0.693563 0.768806
4 0.368064 0.343812 0.319906 0.295919 0.331354 0.597782 0.69265 0.769725
5 0.31995 0.283814 0.248357 0.271183 0.294994 0.320049 0.691742 0.76881
6 0.24875 0.200798 0.212631 0.224459 0.236161 0.24891 0 1
7 0.15434 0.153216 0.153224 0.154128 0.153211 0.154125 0 0

Probability of Tie Win

Banker
Total
Player Total
0 1 2 3 4 5 6 7
0 0.144414 0.094811 0.094811 0.094811 0.094811 0.094608 0.076263 0.076268
1 0.094811 0.146141 0.094762 0.094968 0.094971 0.094967 0.077068 0.077066
2 0.094811 0.094762 0.146139 0.09497 0.094762 0.094968 0.077064 0.077068
3 0.08884 0.089015 0.089018 0.140375 0.089015 0.148221 0.076166 0.077066
4 0.053272 0.053368 0.05309 0.112503 0.342591 0.112503 0.077073 0.077065
5 0.041335 0.041531 0.100738 0.100667 0.100595 0.331064 0.077065 0.077066
6 0.029391 0.088833 0.08883 0.087932 0.088838 0.088759 1 0
7 0.076268 0.077066 0.077068 0.077066 0.077065 0.077066 0 1

After Five Cards

The next three tables display the probability of each possible final outcome given the totals after the first five cards are dealt, assuming a sixth card is forced, based on an 8-deck game. A blank cell means the situation is impossible.

Probability of Banker Win after Five Cards

Banker
Total
Player Total
0 1 2 3 4 5 6 7 8 9
0 0.695236 0.617601 0.540571 0.463553 0.386527 0.309497 0.232147 0.154764 0.077381 0.000000
1 0.922908 0.615304 0.539289 0.462377 0.385459 0.308293 0.231249 0.153977 0.0767 0.000000
2 0.922908 0.845973 0.538302 0.461377 0.385109 0.307946 0.230743 0.153613 0.076905 0.000000
3 0.922922 0.845742 0.768763 0.460953 0.383862 0.306687 0.230225 0.153673 0.076766 0.000000
4 0.923557 0.847902 0.768986 0.690731 0.382642 0.305417 0.229293 0.152621 0.076573 0.000000
5 0.924744 0.84808 0.772494 0.614114 0.307938 0.230948 0.153838 0.076427 0.000000
6 0.923211 0.846428 0.770093 0.232063 0.154635 0.077253 0.000000
7
8
9

Probability of Player Win after Five Cards

Banker
Total
Player Total
0 1 2 3 4 5 6 7 8 9
0 0.000000 0.305793 0.382827 0.459844 0.536872 0.61389 0.691042 0.768423 0.845811 0.92319
1 0.000000 0.077013 0.384686 0.460702 0.537624 0.614788 0.691983 0.769115 0.846393 0.923094
2 0.000000 0.077011 0.154018 0.461692 0.538869 0.615134 0.69249 0.769616 0.846178 0.92309
3 0.000000 0.076989 0.154252 0.231517 0.539219 0.616392 0.693904 0.769562 0.846655 0.923153
4 0.000000 0.076032 0.153694 0.231953 0.309183 0.617492 0.694015 0.77061 0.847738 0.923583
5 0.000000 0.075882 0.152675 0.30853 0.384204 0.692008 0.769077 0.846555 0.923586
6 0.000000 0.076783 0.153579 0.46082 0.768408 0.845958 0.9232
7
8
9

Probability of Tie Win after Five Cards

Banker
Total
Player Total
0 1 2 3 4 5 6 7 8 9
0 0.304764 0.076605 0.076603 0.076604 0.076601 0.076613 0.076811 0.076812 0.076809 0.07681
1 0.077092 0.307683 0.076025 0.076921 0.076917 0.076919 0.076768 0.076908 0.076907 0.076906
2 0.077092 0.077016 0.30768 0.076931 0.076022 0.07692 0.076767 0.076771 0.076918 0.07691
3 0.077078 0.077269 0.076986 0.30753 0.076919 0.076921 0.07587 0.076766 0.076579 0.076847
4 0.076443 0.076065 0.07732 0.077316 0.308175 0.07709 0.076692 0.076769 0.075688 0.076417
5 0.075256 0.076039 0.074831 0.077356 0.307858 0.077044 0.077085 0.077018 0.076414
6 0.076789 0.076789 0.076329 0.307118 0.076956 0.076789 0.0768
7
8
9


Go back to baccarat
Go to baccarat appendix 1
Go to baccarat appendix 2
Go to baccarat appendix 4
Go to baccarat appendix 5
Go to baccarat appendix 6
Go to baccarat appendix 7


Written by: Michael Shackleford

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