Texas Hold'em Bonus Poker - Analysis
House Advantage
The house advantage with the player using the optimal strategy is 2.036921% of the Ante bet wagered. For every $100 wagered on the Ante bet the player will lose $2.04 on average.
The house advantage was determined by exhaustive combinatorial analysis, cycling through every possible combination of two player cards, two dealer cards and five board cards with the player using optimal strategy at each decision point. This took approximately 6 hours on a 3GHz Pentium with 1GB of memory.
Optimal Strategy
A) Flop Bet The player should make the Flop bet when the total expected value of the Ante, Flop, Turn and River bets is greater than that of folding (i.e. -1.000000). From Table 1 below it can be seen that player should make the Flop bet with all hands except for 2-3 offsuit, 2-4 offsuit, 2-5 offsuit, 2-6 offsuit and 2-7 offsuit.
B) Turn and River Bets The player should make the Turn or River bet if the chance of winning the hand is greater than 50% given the player's two cards and the visible board cards. If the chance of winning is less than 50% the player should check instead.
Given the number of different combinations available and the fact the optimal strategy is not readily simplified it is very difficult to list the correct way to play every hand without having a extremely long list of rules and exceptions. Our online optimal strategy calculator provides the correct way to play a hand given the player's cards and visible board cards.
Table 1: Expected Value By Hand
| Hand | Probability | Ante EV | Flop EV | Turn EV | River EV | Total EV |
| 2,3 unsuited | 0.009050 | -0.575363 | -0.707871 | 0.052974 | 0.074333 | -1.155927 |
| 2,4 unsuited | 0.009050 | -0.561881 | -0.672010 | 0.059480 | 0.082806 | -1.091604 |
| 2,5 unsuited | 0.009050 | -0.547147 | -0.628614 | 0.065461 | 0.090313 | -1.019987 |
| 2,6 unsuited | 0.009050 | -0.565779 | -0.636994 | 0.067257 | 0.088793 | -1.046723 |
| 2,7 unsuited | 0.009050 | -0.573922 | -0.616654 | 0.069541 | 0.089735 | -1.031301 |
| 2,8 unsuited | 0.009050 | -0.549834 | -0.526893 | 0.079129 | 0.101595 | -0.896002 |
| 2,9 unsuited | 0.009050 | -0.528389 | -0.436083 | 0.088435 | 0.112905 | -0.763132 |
| 2,T unsuited | 0.009050 | -0.502716 | -0.333266 | 0.097500 | 0.124502 | -0.613979 |
| 2,J unsuited | 0.009050 | -0.480122 | -0.226061 | 0.108999 | 0.137770 | -0.459414 |
| 2,Q unsuited | 0.009050 | -0.454685 | -0.108183 | 0.121235 | 0.151943 | -0.289690 |
| 2,K unsuited | 0.009050 | -0.426432 | 0.020349 | 0.135607 | 0.167808 | -0.102668 |
| 2,A unsuited | 0.009050 | -0.371337 | 0.197142 | 0.163064 | 0.195504 | 0.184373 |
| 3,4 unsuited | 0.009050 | -0.526600 | -0.594164 | 0.071454 | 0.099277 | -0.950034 |
| 3,5 unsuited | 0.009050 | -0.510877 | -0.549409 | 0.077654 | 0.108177 | -0.874456 |
| 3,6 unsuited | 0.009050 | -0.529239 | -0.556895 | 0.079727 | 0.106774 | -0.899632 |
| 3,7 unsuited | 0.009050 | -0.537618 | -0.535910 | 0.081799 | 0.106328 | -0.885400 |
| 3,8 unsuited | 0.009050 | -0.542305 | -0.500647 | 0.084486 | 0.107859 | -0.850607 |
| 3,9 unsuited | 0.009050 | -0.515398 | -0.399219 | 0.094397 | 0.120723 | -0.699497 |
| 3,T unsuited | 0.009050 | -0.489684 | -0.296218 | 0.103483 | 0.132471 | -0.549947 |
| 3,J unsuited | 0.009050 | -0.467087 | -0.188978 | 0.115000 | 0.145858 | -0.395206 |
| 3,Q unsuited | 0.009050 | -0.441686 | -0.071223 | 0.127278 | 0.160068 | -0.225563 |
| 3,K unsuited | 0.009050 | -0.413509 | 0.057028 | 0.141954 | 0.176028 | -0.038499 |
| 3,A unsuited | 0.009050 | -0.357896 | 0.233784 | 0.172954 | 0.205595 | 0.254437 |
| 4,5 unsuited | 0.009050 | -0.476330 | -0.473789 | 0.089428 | 0.125564 | -0.735126 |
| 4,6 unsuited | 0.009050 | -0.493562 | -0.479580 | 0.091868 | 0.125179 | -0.756096 |
| 4,7 unsuited | 0.009050 | -0.501720 | -0.458007 | 0.094060 | 0.125105 | -0.740561 |
| 4,8 unsuited | 0.009050 | -0.506646 | -0.422128 | 0.096506 | 0.125468 | -0.706800 |
| 4,9 unsuited | 0.009050 | -0.507923 | -0.373158 | 0.099502 | 0.127185 | -0.654394 |
| 4,T unsuited | 0.009050 | -0.476840 | -0.259836 | 0.109216 | 0.140423 | -0.487036 |
| 4,J unsuited | 0.009050 | -0.454234 | -0.152545 | 0.120777 | 0.153769 | -0.332233 |
| 4,Q unsuited | 0.009050 | -0.428865 | -0.034893 | 0.133129 | 0.168017 | -0.162612 |
| 4,K unsuited | 0.009050 | -0.400757 | 0.093099 | 0.148282 | 0.184175 | 0.024799 |
| 4,A unsuited | 0.009050 | -0.345173 | 0.269187 | 0.182456 | 0.215171 | 0.321642 |
| 5,6 unsuited | 0.009050 | -0.459170 | -0.402228 | 0.103766 | 0.143314 | -0.614318 |
| 5,7 unsuited | 0.009050 | -0.466277 | -0.379521 | 0.106275 | 0.144042 | -0.595481 |
| 5,8 unsuited | 0.009050 | -0.470990 | -0.342899 | 0.108792 | 0.144610 | -0.560487 |
| 5,9 unsuited | 0.009050 | -0.472500 | -0.293234 | 0.111649 | 0.145492 | -0.508593 |
| 5,T unsuited | 0.009050 | -0.468922 | -0.229962 | 0.114475 | 0.146902 | -0.437507 |
| 5,J unsuited | 0.009050 | -0.441070 | -0.112764 | 0.126719 | 0.161752 | -0.265364 |
| 5,Q unsuited | 0.009050 | -0.415726 | 0.004803 | 0.139238 | 0.176043 | -0.095642 |
| 5,K unsuited | 0.009050 | -0.387682 | 0.132559 | 0.155028 | 0.192462 | 0.092367 |
| 5,A unsuited | 0.009050 | -0.332275 | 0.307861 | 0.192485 | 0.224822 | 0.392893 |
| 6,7 unsuited | 0.009050 | -0.437330 | -0.307090 | 0.121136 | 0.165558 | -0.457726 |
| 6,8 unsuited | 0.009050 | -0.441237 | -0.270364 | 0.123931 | 0.166656 | -0.421014 |
| 6,9 unsuited | 0.009050 | -0.442651 | -0.220346 | 0.126870 | 0.167737 | -0.368390 |
| 6,T unsuited | 0.009050 | -0.439328 | -0.156320 | 0.129681 | 0.168709 | -0.297258 |
| 6,J unsuited | 0.009050 | -0.438181 | -0.086229 | 0.134987 | 0.171449 | -0.217975 |
| 6,Q unsuited | 0.009050 | -0.407679 | 0.040962 | 0.148245 | 0.187257 | -0.031216 |
| 6,K unsuited | 0.009050 | -0.379585 | 0.168931 | 0.163840 | 0.203633 | 0.156819 |
| 6,A unsuited | 0.009050 | -0.349001 | 0.307298 | 0.188795 | 0.223627 | 0.370719 |
| 7,8 unsuited | 0.009050 | -0.412493 | -0.197968 | 0.138525 | 0.187897 | -0.284039 |
| 7,9 unsuited | 0.009050 | -0.413180 | -0.148088 | 0.141798 | 0.189483 | -0.229986 |
| 7,T unsuited | 0.009050 | -0.409739 | -0.083675 | 0.144960 | 0.190685 | -0.157768 |
| 7,J unsuited | 0.009050 | -0.408885 | -0.012723 | 0.150207 | 0.193184 | -0.078216 |
| 7,Q unsuited | 0.009050 | -0.404225 | 0.070627 | 0.156632 | 0.196854 | 0.019887 |
| 7,K unsuited | 0.009050 | -0.371174 | 0.207494 | 0.173445 | 0.215147 | 0.224911 |
| 7,A unsuited | 0.009050 | -0.335821 | 0.353648 | 0.200986 | 0.237986 | 0.456798 |
| 8,9 unsuited | 0.009050 | -0.384650 | -0.076119 | 0.157403 | 0.210989 | -0.092377 |
| 8,T unsuited | 0.009050 | -0.380172 | -0.011149 | 0.162005 | 0.212934 | -0.016383 |
| 8,J unsuited | 0.009050 | -0.379517 | 0.059607 | 0.167163 | 0.215456 | 0.062709 |
| 8,Q unsuited | 0.009050 | -0.375140 | 0.143992 | 0.173373 | 0.219215 | 0.161440 |
| 8,K unsuited | 0.009050 | -0.366965 | 0.240807 | 0.183168 | 0.225799 | 0.282809 |
| 8,A unsuited | 0.009050 | -0.326912 | 0.394904 | 0.213697 | 0.252027 | 0.533716 |
| 9,T unsuited | 0.009050 | -0.350736 | 0.061267 | 0.180957 | 0.235539 | 0.127027 |
| 9,J unsuited | 0.009050 | -0.350366 | 0.130048 | 0.186644 | 0.238014 | 0.204340 |
| 9,Q unsuited | 0.009050 | -0.346164 | 0.214417 | 0.194072 | 0.242139 | 0.304465 |
| 9,K unsuited | 0.009050 | -0.338226 | 0.312477 | 0.204492 | 0.249445 | 0.428187 |
| 9,A unsuited | 0.009050 | -0.321969 | 0.430912 | 0.227818 | 0.265646 | 0.602407 |
| T,J unsuited | 0.009050 | -0.314276 | 0.209908 | 0.210738 | 0.263909 | 0.370279 |
| T,Q unsuited | 0.009050 | -0.310401 | 0.291631 | 0.221678 | 0.268785 | 0.471694 |
| T,K unsuited | 0.009050 | -0.302668 | 0.389557 | 0.235631 | 0.277709 | 0.600228 |
| T,A unsuited | 0.009050 | -0.286830 | 0.508866 | 0.263517 | 0.296387 | 0.781940 |
| J,Q unsuited | 0.009050 | -0.306044 | 0.325388 | 0.234055 | 0.282156 | 0.535555 |
| J,K unsuited | 0.009050 | -0.298475 | 0.422747 | 0.249438 | 0.292050 | 0.665761 |
| J,A unsuited | 0.009050 | -0.282602 | 0.542530 | 0.279916 | 0.312649 | 0.852493 |
| Q,K unsuited | 0.009050 | -0.293571 | 0.458232 | 0.263507 | 0.307276 | 0.735444 |
| Q,A unsuited | 0.009050 | -0.277894 | 0.577274 | 0.296528 | 0.329404 | 0.925311 |
| K,A unsuited | 0.009050 | -0.272659 | 0.612803 | 0.313330 | 0.345842 | 0.999317 |
| 2,3 suited | 0.003017 | -0.495875 | -0.560623 | 0.066692 | 0.102820 | -0.886985 |
| 2,4 suited | 0.003017 | -0.483298 | -0.526839 | 0.073337 | 0.110792 | -0.826008 |
| 2,5 suited | 0.003017 | -0.469516 | -0.486027 | 0.079559 | 0.118004 | -0.757980 |
| 2,6 suited | 0.003017 | -0.486835 | -0.493241 | 0.080523 | 0.116672 | -0.782881 |
| 2,7 suited | 0.003017 | -0.494377 | -0.473764 | 0.082495 | 0.117570 | -0.768076 |
| 2,8 suited | 0.003017 | -0.471643 | -0.389135 | 0.092597 | 0.129160 | -0.639021 |
| 2,9 suited | 0.003017 | -0.451312 | -0.303393 | 0.104300 | 0.140255 | -0.510150 |
| 2,T suited | 0.003017 | -0.427016 | -0.206421 | 0.117067 | 0.151774 | -0.364595 |
| 2,J suited | 0.003017 | -0.405416 | -0.104874 | 0.132236 | 0.165345 | -0.212709 |
| 2,Q suited | 0.003017 | -0.381105 | 0.006761 | 0.148597 | 0.180799 | -0.044946 |
| 2,K suited | 0.003017 | -0.354105 | 0.128469 | 0.167425 | 0.199352 | 0.141142 |
| 2,A suited | 0.003017 | -0.302002 | 0.295156 | 0.200743 | 0.231125 | 0.425022 |
| 3,4 suited | 0.003017 | -0.450565 | -0.454322 | 0.085668 | 0.126174 | -0.693044 |
| 3,5 suited | 0.003017 | -0.435876 | -0.412282 | 0.092282 | 0.134645 | -0.621230 |
| 3,6 suited | 0.003017 | -0.452945 | -0.418658 | 0.093464 | 0.133362 | -0.644776 |
| 3,7 suited | 0.003017 | -0.460712 | -0.398563 | 0.095062 | 0.133131 | -0.631082 |
| 3,8 suited | 0.003017 | -0.464978 | -0.365060 | 0.097855 | 0.134706 | -0.597478 |
| 3,9 suited | 0.003017 | -0.439566 | -0.269430 | 0.110831 | 0.147282 | -0.450882 |
| 3,T suited | 0.003017 | -0.415236 | -0.172300 | 0.123689 | 0.158929 | -0.304918 |
| 3,J suited | 0.003017 | -0.393638 | -0.070735 | 0.138887 | 0.172618 | -0.152868 |
| 3,Q suited | 0.003017 | -0.369365 | 0.040770 | 0.155264 | 0.188221 | 0.014889 |
| 3,K suited | 0.003017 | -0.342441 | 0.162199 | 0.174368 | 0.207249 | 0.201376 |
| 3,A suited | 0.003017 | -0.289867 | 0.328813 | 0.210581 | 0.241091 | 0.490618 |
| 4,5 suited | 0.003017 | -0.403827 | -0.341863 | 0.104569 | 0.150923 | -0.490199 |
| 4,6 suited | 0.003017 | -0.419851 | -0.346667 | 0.106233 | 0.150578 | -0.509707 |
| 4,7 suited | 0.003017 | -0.427416 | -0.326028 | 0.107986 | 0.150604 | -0.494853 |
| 4,8 suited | 0.003017 | -0.431911 | -0.291935 | 0.111018 | 0.151193 | -0.461634 |
| 4,9 suited | 0.003017 | -0.432941 | -0.245521 | 0.116254 | 0.153077 | -0.409130 |
| 4,T suited | 0.003017 | -0.403616 | -0.138780 | 0.130127 | 0.166162 | -0.246108 |
| 4,J suited | 0.003017 | -0.382016 | -0.037182 | 0.145339 | 0.179834 | -0.094025 |
| 4,Q suited | 0.003017 | -0.357776 | 0.074212 | 0.161811 | 0.195601 | 0.073849 |
| 4,K suited | 0.003017 | -0.330921 | 0.195385 | 0.181284 | 0.215180 | 0.260929 |
| 4,A suited | 0.003017 | -0.278384 | 0.361345 | 0.219981 | 0.250612 | 0.553554 |
| 5,6 suited | 0.003017 | -0.387949 | -0.274665 | 0.118828 | 0.167595 | -0.376191 |
| 5,7 suited | 0.003017 | -0.394542 | -0.252979 | 0.121139 | 0.168392 | -0.357989 |
| 5,8 suited | 0.003017 | -0.398843 | -0.218200 | 0.124912 | 0.169113 | -0.323018 |
| 5,9 suited | 0.003017 | -0.400096 | -0.171125 | 0.130142 | 0.170246 | -0.270833 |
| 5,T suited | 0.003017 | -0.396576 | -0.111350 | 0.135886 | 0.172066 | -0.199974 |
| 5,J suited | 0.003017 | -0.370095 | -0.000526 | 0.152058 | 0.187190 | -0.031373 |
| 5,Q suited | 0.003017 | -0.345883 | 0.110776 | 0.168680 | 0.203172 | 0.136745 |
| 5,K suited | 0.003017 | -0.319092 | 0.231717 | 0.188700 | 0.223468 | 0.324793 |
| 5,A suited | 0.003017 | -0.266729 | 0.396917 | 0.230101 | 0.260477 | 0.620767 |
| 6,7 suited | 0.003017 | -0.367588 | -0.185129 | 0.136894 | 0.188752 | -0.227072 |
| 6,8 suited | 0.003017 | -0.371145 | -0.150269 | 0.141156 | 0.189981 | -0.190276 |
| 6,9 suited | 0.003017 | -0.372311 | -0.102868 | 0.146458 | 0.191257 | -0.137464 |
| 6,T suited | 0.003017 | -0.369033 | -0.042373 | 0.152203 | 0.192660 | -0.066542 |
| 6,J suited | 0.003017 | -0.367602 | 0.024236 | 0.160373 | 0.196266 | 0.013273 |
| 6,Q suited | 0.003017 | -0.338593 | 0.144503 | 0.178011 | 0.213707 | 0.197628 |
| 6,K suited | 0.003017 | -0.311758 | 0.265629 | 0.197912 | 0.234074 | 0.385858 |
| 6,A suited | 0.003017 | -0.282627 | 0.396233 | 0.226820 | 0.259394 | 0.599820 |
| 7,8 suited | 0.003017 | -0.344413 | -0.082546 | 0.156958 | 0.210141 | -0.059860 |
| 7,9 suited | 0.003017 | -0.344908 | -0.035291 | 0.162731 | 0.211917 | -0.005551 |
| 7,T suited | 0.003017 | -0.341522 | 0.025562 | 0.168626 | 0.213622 | 0.066287 |
| 7,J suited | 0.003017 | -0.340368 | 0.092991 | 0.176585 | 0.217059 | 0.146267 |
| 7,Q suited | 0.003017 | -0.335599 | 0.172091 | 0.186494 | 0.223038 | 0.246024 |
| 7,K suited | 0.003017 | -0.304152 | 0.301510 | 0.208069 | 0.245680 | 0.451106 |
| 7,A suited | 0.003017 | -0.270593 | 0.439358 | 0.240248 | 0.273897 | 0.682911 |
| 8,9 suited | 0.003017 | -0.318387 | 0.032030 | 0.179558 | 0.232447 | 0.125648 |
| 8,T suited | 0.003017 | -0.314043 | 0.093375 | 0.186486 | 0.235137 | 0.200955 |
| 8,J suited | 0.003017 | -0.313074 | 0.160626 | 0.194375 | 0.238868 | 0.280795 |
| 8,Q suited | 0.003017 | -0.308571 | 0.240709 | 0.204009 | 0.245186 | 0.381333 |
| 8,K suited | 0.003017 | -0.300458 | 0.332494 | 0.218151 | 0.256374 | 0.506560 |
| 8,A suited | 0.003017 | -0.262525 | 0.477752 | 0.254133 | 0.287859 | 0.757219 |
| 9,T suited | 0.003017 | -0.286695 | 0.161101 | 0.205896 | 0.257269 | 0.337570 |
| 9,J suited | 0.003017 | -0.285995 | 0.226499 | 0.214395 | 0.261368 | 0.416267 |
| 9,Q suited | 0.003017 | -0.281655 | 0.306573 | 0.225201 | 0.268289 | 0.518407 |
| 9,K suited | 0.003017 | -0.273765 | 0.399539 | 0.240197 | 0.280175 | 0.646145 |
| 9,A suited | 0.003017 | -0.258152 | 0.511249 | 0.268126 | 0.301053 | 0.822276 |
| T,J suited | 0.003017 | -0.252472 | 0.301114 | 0.238738 | 0.287154 | 0.574533 |
| T,Q suited | 0.003017 | -0.248441 | 0.378702 | 0.252154 | 0.294602 | 0.677017 |
| T,K suited | 0.003017 | -0.240744 | 0.471542 | 0.269958 | 0.307764 | 0.808521 |
| T,A suited | 0.003017 | -0.225527 | 0.584095 | 0.301241 | 0.330479 | 0.990289 |
| J,Q suited | 0.003017 | -0.244560 | 0.410368 | 0.264231 | 0.307627 | 0.737666 |
| J,K suited | 0.003017 | -0.237013 | 0.502694 | 0.283266 | 0.321545 | 0.870492 |
| J,A suited | 0.003017 | -0.221757 | 0.615707 | 0.316538 | 0.345722 | 1.056211 |
| Q,K suited | 0.003017 | -0.232619 | 0.536016 | 0.297005 | 0.336103 | 0.936506 |
| Q,A suited | 0.003017 | -0.217543 | 0.648354 | 0.332155 | 0.361466 | 1.124432 |
| K,A suited | 0.003017 | -0.212834 | 0.681785 | 0.347988 | 0.377085 | 1.194024 |
| 2,2 pair | 0.004525 | -0.386010 | 0.013361 | 0.101812 | 0.136700 | -0.134137 |
| 3,3 pair | 0.004525 | -0.348393 | 0.147723 | 0.123606 | 0.155649 | 0.078584 |
| 4,4 pair | 0.004525 | -0.311038 | 0.280913 | 0.163053 | 0.189360 | 0.322289 |
| 5,5 pair | 0.004525 | -0.273943 | 0.412997 | 0.215848 | 0.234554 | 0.589457 |
| 6,6 pair | 0.004525 | -0.244036 | 0.531390 | 0.269262 | 0.280849 | 0.837465 |
| 7,7 pair | 0.004525 | -0.213974 | 0.649441 | 0.325708 | 0.332494 | 1.093669 |
| 8,8 pair | 0.004525 | -0.184073 | 0.766521 | 0.383477 | 0.387123 | 1.353048 |
| 9,9 pair | 0.004525 | -0.154389 | 0.882290 | 0.441200 | 0.443013 | 1.612114 |
| T,T pair | 0.004525 | -0.122761 | 1.000471 | 0.500236 | 0.501044 | 1.878990 |
| J,J pair | 0.004525 | -0.100759 | 1.098779 | 0.549389 | 0.550006 | 2.097415 |
| Q,Q pair | 0.004525 | -0.078647 | 1.197007 | 0.598503 | 0.598976 | 2.315838 |
| K,K pair | 0.004525 | -0.056288 | 1.295827 | 0.647914 | 0.648239 | 2.535691 |
| A,A pair | 0.004525 | -0.027555 | 1.408149 | 0.704074 | 0.704230 | 2.788899 |
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