Blackjack Probability Worksheet
- Appendices
- Miscellaneous
- External Links
- Blackjack Probability Theory
- Blackjack Probability Math
- Blackjack Math And Probabilities
- Blackjack Probability To Win
Introduction
The following tables display expected returns for any play in blackjack based on the following rules: dealer stands on a soft 17, an infinite deck, the player may double after a split, split up to three times except for aces, and draw only one card to split aces. Based on these rules, the player's expected value is -0.511734%.
- Learners explore the mathematical probabilities involved in gambling and how these factors affect people's behavior. They work in pairs and conduct and experiment pertaining to blackjack. The class creates a graph showing the trends found.
- Blackjack and Probability Chongwu Ruan Math 190S-Hubert Bray July 24, 2017 1 Introduction Blackjack is an usual game in gambling house and to beat the dealer and make money, people have done lots of research on it. They come up with several basic strategy which is consist of three tables corresponding to the different rules.
To use this table, look up the returns for any given play, the one with the greatest return is the best play. For example suppose you have two 8's and the dealer has a 10. The return by standing is -0.5404, by hitting is -0.5398, doubling is -1.0797, and by splitting is -0.4807. So splitting 8's you stand to lose the least, 48.07 cents per original dollar bet, and is thus the best play.
Stand
Player's Expected Return by StandingExpand
By the inclusion/exclusion principle, the probability of at least one getting blackjack is the sum of probabilities of each individual getting blackjack minus the probability of both getting blackjack (removes redundant overlap): P either or both = 2 P blackjack − P both.
| Player's Hand | Dealer's Up Card | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Ace | |
| 0-16 | -0.292784 | -0.252250 | -0.211063 | -0.167193 | -0.153699 | -0.475375 | -0.510518 | -0.543150 | -0.540430 | -0.666951 |
| 17 | -0.152975 | -0.117216 | -0.080573 | -0.044941 | 0.011739 | -0.106809 | -0.381951 | -0.423154 | -0.419721 | -0.478033 |
| 18 | 0.121742 | 0.148300 | 0.175854 | 0.199561 | 0.283444 | 0.399554 | 0.105951 | -0.183163 | -0.178301 | -0.100199 |
| 19 | 0.386305 | 0.404363 | 0.423179 | 0.439512 | 0.495977 | 0.615976 | 0.593854 | 0.287597 | 0.063118 | 0.277636 |
| 20 | 0.639987 | 0.650272 | 0.661050 | 0.670360 | 0.703959 | 0.773227 | 0.791815 | 0.758357 | 0.554538 | 0.655470 |
| 21 | 0.882007 | 0.885300 | 0.888767 | 0.891754 | 0.902837 | 0.925926 | 0.930605 | 0.939176 | 0.962624 | 0.922194 |
Hit
Player's Expected Return by HittingExpand
| Player's Hand | Dealer's Up Card | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Ace | |
| 4 | -0.114913 | -0.082613 | -0.049367 | -0.012380 | 0.011130 | -0.088279 | -0.159334 | -0.240666 | -0.289198 | -0.253077 |
| 5 | -0.128216 | -0.095310 | -0.061479 | -0.023979 | -0.001186 | -0.119447 | -0.188093 | -0.266615 | -0.313412 | -0.278575 |
| 6 | -0.140759 | -0.107291 | -0.072917 | -0.034916 | -0.013006 | -0.151933 | -0.217242 | -0.292641 | -0.337749 | -0.304147 |
| 7 | -0.109183 | -0.076583 | -0.043022 | -0.007271 | 0.029185 | -0.068808 | -0.210605 | -0.285365 | -0.319055 | -0.310072 |
| 8 | -0.021798 | 0.008005 | 0.038784 | 0.070805 | 0.114960 | 0.082207 | -0.059898 | -0.210186 | -0.249375 | -0.197029 |
| 9 | 0.074446 | 0.101265 | 0.128981 | 0.158032 | 0.196019 | 0.171868 | 0.098376 | -0.052178 | -0.152953 | -0.065681 |
| 10 | 0.182500 | 0.206088 | 0.230470 | 0.256259 | 0.287795 | 0.256909 | 0.197954 | 0.116530 | 0.025309 | 0.081450 |
| 11 | 0.238351 | 0.260325 | 0.283020 | 0.307350 | 0.333690 | 0.292147 | 0.229982 | 0.158257 | 0.119482 | 0.143001 |
| 12 | -0.253390 | -0.233691 | -0.213537 | -0.193271 | -0.170526 | -0.212848 | -0.271575 | -0.340013 | -0.381043 | -0.350540 |
| 13 | -0.307791 | -0.291210 | -0.274224 | -0.257333 | -0.235626 | -0.269073 | -0.323605 | -0.387155 | -0.425254 | -0.396930 |
| 14 | -0.362192 | -0.348729 | -0.334911 | -0.321395 | -0.300726 | -0.321282 | -0.371919 | -0.430930 | -0.466307 | -0.440007 |
| 15 | -0.416594 | -0.406249 | -0.395599 | -0.385457 | -0.365826 | -0.369762 | -0.416782 | -0.471578 | -0.504428 | -0.480006 |
| 16 | -0.470995 | -0.463768 | -0.456286 | -0.449520 | -0.430927 | -0.414779 | -0.458440 | -0.509322 | -0.539826 | -0.517149 |
| 17 | -0.536151 | -0.531674 | -0.527011 | -0.522986 | -0.508753 | -0.483486 | -0.505983 | -0.553695 | -0.584463 | -0.557300 |
| 18 | -0.622439 | -0.620005 | -0.617462 | -0.615260 | -0.607479 | -0.591144 | -0.591056 | -0.616528 | -0.647671 | -0.626515 |
| 19 | -0.729077 | -0.728033 | -0.726937 | -0.725991 | -0.722554 | -0.715450 | -0.713660 | -0.715574 | -0.729449 | -0.724795 |
| 20 | -0.855230 | -0.854977 | -0.854710 | -0.854480 | -0.853628 | -0.851852 | -0.851492 | -0.850833 | -0.849029 | -0.852139 |
| Soft 12 | 0.081836 | 0.103507 | 0.126596 | 0.156482 | 0.185954 | 0.165473 | 0.095115 | 0.000066 | -0.070002 | -0.020478 |
| Soft 13 | 0.046636 | 0.074119 | 0.102477 | 0.133363 | 0.161693 | 0.122386 | 0.054057 | -0.037695 | -0.104851 | -0.057308 |
| Soft 14 | 0.022392 | 0.050807 | 0.080081 | 0.111894 | 0.139165 | 0.079507 | 0.013277 | -0.075163 | -0.139467 | -0.093874 |
| Soft 15 | -0.000121 | 0.029160 | 0.059285 | 0.091960 | 0.118246 | 0.037028 | -0.027055 | -0.112189 | -0.173704 | -0.130027 |
| Soft 16 | -0.021025 | 0.009059 | 0.039975 | 0.073449 | 0.098821 | -0.004890 | -0.066795 | -0.148644 | -0.207441 | -0.165637 |
| Soft 17 | -0.000491 | 0.028975 | 0.059326 | 0.091189 | 0.128052 | 0.053823 | -0.072915 | -0.149787 | -0.196867 | -0.179569 |
| Soft 18 | 0.062905 | 0.090248 | 0.118502 | 0.147613 | 0.190753 | 0.170676 | 0.039677 | -0.100744 | -0.143808 | -0.092935 |
| Soft 19 | 0.123958 | 0.149340 | 0.175577 | 0.202986 | 0.239799 | 0.220620 | 0.152270 | 0.007893 | -0.088096 | -0.005743 |
| Soft 20 | 0.182500 | 0.206088 | 0.230470 | 0.256259 | 0.287795 | 0.256909 | 0.197954 | 0.116530 | 0.025309 | 0.081450 |
| Soft 21 | 0.238351 | 0.260325 | 0.283020 | 0.307350 | 0.333690 | 0.292147 | 0.229982 | 0.158257 | 0.119482 | 0.143001 |
Double
Player's Expected Return by DoublingExpand
| Player's Hand | Dealer's Up Card | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Ace | |
| Hard 4 | -0.585567 | -0.504500 | -0.422126 | -0.334385 | -0.307398 | -0.950750 | -1.021035 | -1.086299 | -1.080861 | -1.333902 |
| Hard 5 | -0.585567 | -0.504500 | -0.422126 | -0.334385 | -0.307398 | -0.950750 | -1.021035 | -1.086299 | -1.080861 | -1.333902 |
| Hard 6 | -0.564058 | -0.483726 | -0.402051 | -0.315577 | -0.281946 | -0.894048 | -1.001256 | -1.067839 | -1.062290 | -1.304837 |
| Hard 7 | -0.435758 | -0.359779 | -0.282299 | -0.202730 | -0.138337 | -0.589336 | -0.847076 | -0.957074 | -0.950866 | -1.130452 |
| Hard 8 | -0.204491 | -0.136216 | -0.066372 | 0.003456 | 0.087015 | -0.187730 | -0.451987 | -0.718501 | -0.746588 | -0.810746 |
| Hard 9 | 0.061119 | 0.120816 | 0.181949 | 0.243057 | 0.317055 | 0.104250 | -0.026442 | -0.300996 | -0.466707 | -0.432911 |
| Hard 10 | 0.358939 | 0.409321 | 0.460940 | 0.512517 | 0.575590 | 0.392412 | 0.286636 | 0.144328 | -0.008659 | -0.014042 |
| Hard 11 | 0.470641 | 0.517795 | 0.566041 | 0.614699 | 0.667380 | 0.462889 | 0.350693 | 0.227783 | 0.179689 | 0.109061 |
| Hard 12 | -0.506780 | -0.467382 | -0.427073 | -0.386542 | -0.341052 | -0.506712 | -0.615661 | -0.737506 | -0.796841 | -0.829344 |
| Hard 13 | -0.615582 | -0.582420 | -0.548448 | -0.514667 | -0.471253 | -0.587423 | -0.690966 | -0.807790 | -0.867544 | -0.880582 |
| Hard 14 | -0.724385 | -0.697459 | -0.669823 | -0.642791 | -0.601453 | -0.668135 | -0.766271 | -0.878075 | -0.938247 | -0.931821 |
| Hard 15 | -0.833187 | -0.812497 | -0.791198 | -0.770915 | -0.731653 | -0.748846 | -0.841576 | -0.948360 | -1.008950 | -0.983059 |
| Hard 16 | -0.941990 | -0.927536 | -0.912573 | -0.899039 | -0.861853 | -0.829558 | -0.916881 | -1.018644 | -1.079653 | -1.034297 |
| Hard 17 | -1.072302 | -1.063348 | -1.054023 | -1.045971 | -1.017505 | -0.966972 | -1.011965 | -1.107390 | -1.168926 | -1.114600 |
| Hard 18 | -1.244877 | -1.240010 | -1.234924 | -1.230519 | -1.214958 | -1.182288 | -1.182112 | -1.233057 | -1.295342 | -1.253031 |
| Hard 19 | -1.458155 | -1.456066 | -1.453874 | -1.451983 | -1.445108 | -1.430899 | -1.427320 | -1.431149 | -1.458898 | -1.449590 |
| Hard 20 | -1.710461 | -1.709954 | -1.709420 | -1.708961 | -1.707256 | -1.703704 | -1.702984 | -1.701665 | -1.698058 | -1.704278 |
| Soft 12 | -0.071570 | -0.007228 | 0.058427 | 0.125954 | 0.179748 | -0.183866 | -0.314441 | -0.456367 | -0.514028 | -0.624391 |
| Soft 13 | -0.071570 | -0.007228 | 0.058427 | 0.125954 | 0.179748 | -0.183866 | -0.314441 | -0.456367 | -0.514028 | -0.624391 |
| Soft 14 | -0.071570 | -0.007228 | 0.058427 | 0.125954 | 0.179748 | -0.183866 | -0.314441 | -0.456367 | -0.514028 | -0.624391 |
| Soft 15 | -0.071570 | -0.007228 | 0.058427 | 0.125954 | 0.179748 | -0.183866 | -0.314441 | -0.456367 | -0.514028 | -0.624391 |
| Soft 16 | -0.071570 | -0.007228 | 0.058427 | 0.125954 | 0.179748 | -0.183866 | -0.314441 | -0.456367 | -0.514028 | -0.624391 |
| Soft 17 | -0.007043 | 0.055095 | 0.118653 | 0.182378 | 0.256104 | -0.013758 | -0.255102 | -0.400984 | -0.458316 | -0.537198 |
| Soft 18 | 0.119750 | 0.177641 | 0.237004 | 0.295225 | 0.381506 | 0.219948 | -0.029917 | -0.290219 | -0.346892 | -0.362813 |
| Soft 19 | 0.241855 | 0.295824 | 0.351154 | 0.405972 | 0.479599 | 0.319835 | 0.195269 | -0.072946 | -0.235468 | -0.188428 |
| Soft 20 | 0.358939 | 0.409321 | 0.460940 | 0.512517 | 0.575590 | 0.392412 | 0.286636 | 0.144328 | -0.008659 | -0.014042 |
| Soft 21 | 0.470641 | 0.517795 | 0.566041 | 0.614699 | 0.667380 | 0.462889 | 0.350693 | 0.227783 | 0.179689 | 0.109061 |
Split
Player's Expected Return by SplittingExpand
| Player's Hand | Dealer's Up Card | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Ace | |
| 2,2 | -0.084336 | -0.015650 | 0.059088 | 0.151665 | 0.226890 | 0.006743 | -0.176693 | -0.386883 | -0.507175 | -0.433570 |
| 3,3 | -0.137710 | -0.056273 | 0.029932 | 0.126284 | 0.201318 | -0.053043 | -0.231843 | -0.436607 | -0.553507 | -0.482405 |
| 4,4 | -0.192325 | -0.108712 | -0.020395 | 0.081913 | 0.151377 | -0.166452 | -0.326068 | -0.511152 | -0.625044 | -0.560206 |
| 5,5 | -0.290154 | -0.208718 | -0.119335 | -0.019231 | 0.045404 | -0.293928 | -0.454237 | -0.634113 | -0.729969 | -0.668811 |
| 6,6 | -0.212560 | -0.119715 | -0.021320 | 0.080912 | 0.153668 | -0.264427 | -0.425122 | -0.610576 | -0.716103 | -0.653362 |
| 7,7 | -0.131478 | -0.043733 | 0.049255 | 0.146678 | 0.247385 | -0.050148 | -0.391981 | -0.577584 | -0.657268 | -0.651641 |
| 8,8 | 0.073852 | 0.146187 | 0.220849 | 0.297475 | 0.409329 | 0.321042 | -0.022736 | -0.387228 | -0.480686 | -0.372535 |
| 9,9 | 0.195625 | 0.258548 | 0.323474 | 0.391987 | 0.471339 | 0.364837 | 0.234447 | -0.078010 | -0.317336 | -0.136810 |
| 10,10 | 0.134774 | 0.212836 | 0.293403 | 0.380367 | 0.468117 | 0.296633 | 0.064443 | -0.206733 | -0.371278 | -0.249494 |
| A,A | 0.470641 | 0.517795 | 0.566041 | 0.614699 | 0.667380 | 0.462889 | 0.350693 | 0.227783 | 0.179689 | 0.109061 |
Here are basic strategy tables for infinite decks.
The only differences between infinite and four decks is to hit soft 13 vs. 5, and soft 15 vs 4 only when the dealer stands on soft 17.
I have had a lot of requests for my actual spreadsheet through the years. It is available to the public at Google docs. Note that this document allows for infinite re-splitting, while the tables above are based on a maximum of three splits (except aces).
Written by: Michael Shackleford
Thread Rating:
dlarcher10
There is a video blackjack machine that offers a bonus for having 4 blackjacks at the same time (dealer blackjack excluded).
What are the chances of getting 4 blackjacks?
A single deck is used.
We get to play on 7 hands per game.
What are the chances of getting 4 blackjacks?
A single deck is used.
We get to play on 7 hands per game.
Hunterhill
I don't know the odds, but this has to be one of the worst bonuses I've heard of.
RomesThanks for this post from:
Hi dlarcher10, welcome to the forums.
I'd encourage you to 'sound this out' and try to solve it yourself, even though I'm providing the info below. Problems like this sound really complicated, but really aren't when you 'sound it out' and take the probably one piece at a time.
In statistics when you need multiple events to happen (frequently when you use the word 'AND') the resulting probability is multiplicative... When any one of multiple events could happen (frequently when you use the word 'OR') the resulting probability is additive. Thus:
P(4 blackjacks) = P(1st blackjack) * P(2nd blackjack given 1st) * P(3rd blackjack given 1st 2) * P(4th blackjack given 1st 3)
P(1st blackjack) = P(1st ace BJ) OR P(1st 10 BJ) = P(1st ace 2nd 10) + P(1st 10 2nd ace) = [(4/52)*(16/51)] + [(16/52)*(4/51)] = (.0769*.3137) + (.3077*.0784) = .0241 + .0241 = .0482... or ~4.8% chance of getting dealt a blackjack (~1 in 21 hands, as shown previously by the Wiz).
P(2nd blackjack given 1st) = (same as above just remove one ace, 10, and 2 total cards from the deck)... [(3/50)*(15/49)]*2 = (.06*.3061)*2 = .0368, or ~3.7%
P(3rd blackjack given 1st 2) = (same as above with more removals)... [(2/48)*(14/47)]*2 = .0248, or ~2.5%
P(4th blackjack given 1st 3) = (same as above with more removals)... [(1/46)*(13/45)]*2 = .0126, or ~ 1.3%
Thus, P(4 blackjacks) = .0482 * .0368 * .0248 * .0126 = .0000005543, or ~ .00005543%... which is ~1 in 1.9 million.
I'd expect the payout to be a million bucks, and then they'd still be shorting you. So more than likely a terrible bet.
I'd encourage you to 'sound this out' and try to solve it yourself, even though I'm providing the info below. Problems like this sound really complicated, but really aren't when you 'sound it out' and take the probably one piece at a time.
In statistics when you need multiple events to happen (frequently when you use the word 'AND') the resulting probability is multiplicative... When any one of multiple events could happen (frequently when you use the word 'OR') the resulting probability is additive. Thus:
P(4 blackjacks) = P(1st blackjack) * P(2nd blackjack given 1st) * P(3rd blackjack given 1st 2) * P(4th blackjack given 1st 3)
P(1st blackjack) = P(1st ace BJ) OR P(1st 10 BJ) = P(1st ace 2nd 10) + P(1st 10 2nd ace) = [(4/52)*(16/51)] + [(16/52)*(4/51)] = (.0769*.3137) + (.3077*.0784) = .0241 + .0241 = .0482... or ~4.8% chance of getting dealt a blackjack (~1 in 21 hands, as shown previously by the Wiz).
P(2nd blackjack given 1st) = (same as above just remove one ace, 10, and 2 total cards from the deck)... [(3/50)*(15/49)]*2 = (.06*.3061)*2 = .0368, or ~3.7%
P(3rd blackjack given 1st 2) = (same as above with more removals)... [(2/48)*(14/47)]*2 = .0248, or ~2.5%
P(4th blackjack given 1st 3) = (same as above with more removals)... [(1/46)*(13/45)]*2 = .0126, or ~ 1.3%
Thus, P(4 blackjacks) = .0482 * .0368 * .0248 * .0126 = .0000005543, or ~ .00005543%... which is ~1 in 1.9 million.
I'd expect the payout to be a million bucks, and then they'd still be shorting you. So more than likely a terrible bet.
dlarcher10
2 * (16/52 * 4/51) = 128/2652
128/2652 * Number of hands = 128/2652 * 7 = 896/2652 ~ 1/3
Did you take into account that any of the 7 hands can make 4 blackjacks?
ThatDonGuy128/2652 * Number of hands = 128/2652 * 7 = 896/2652 ~ 1/3
Did you take into account that any of the 7 hands can make 4 blackjacks?
Thanks for this post from:
Assuming there are 7 players:
There are (7)C(4) = 35 groups of 4 players that can have the blackjacks
The first player can have any of 4 aces and any of 16 10-cards, or 64 possible hands
The second player can have any of the 3 remaining aces and any of the 15 remaining 10-cards, or 45 hands
The third player can have either of the 2 remaining aces and any of the 14 remaining 10-cards, or 28 hands
The fourth player can have the remaining ace and any of the 13 remaining 10-cards, or 13 hands
The first of the other three players can have any of the (44)C(2) remaining hands, the second any of the (42)C(2) remaining hands, and the third any of the (40)C(2) remaining hands.
Divide this product by (52)C(2) x (50)C(2) x (48)C(2) x (46)C(2) x (44)C(2) x (42)C(2) x (40)C(2), and you get about 1 / 51,685.
Simulation seems to confirm this calculation.
There are (7)C(4) = 35 groups of 4 players that can have the blackjacks
The first player can have any of 4 aces and any of 16 10-cards, or 64 possible hands
The second player can have any of the 3 remaining aces and any of the 15 remaining 10-cards, or 45 hands
The third player can have either of the 2 remaining aces and any of the 14 remaining 10-cards, or 28 hands
The fourth player can have the remaining ace and any of the 13 remaining 10-cards, or 13 hands
The first of the other three players can have any of the (44)C(2) remaining hands, the second any of the (42)C(2) remaining hands, and the third any of the (40)C(2) remaining hands.
Divide this product by (52)C(2) x (50)C(2) x (48)C(2) x (46)C(2) x (44)C(2) x (42)C(2) x (40)C(2), and you get about 1 / 51,685.
Simulation seems to confirm this calculation.
Blackjack Probability Theory
ThatDonGuy
2 * (16/52 * 4/51) = 128/2652
128/2652 * Number of hands = 128/2652 * 7 = 896/2652 ~ 1/3
Did you take into account that any of the 7 hands can make 4 blackjacks?
I think he missed the part where it said they could play up to 7 hands.
If there are only 4 hands, the probability is about 1 / 1,808,900
If there are 5, about 1 / 361,800
If there are 6, about 1 / 120,600
dlarcher10
7 x 2 x 16/52 x 4/51 = 896/2652
6 x 2 x 15/50 x 3/49 = 540/2450
5 x 2 x 14/48 x 2/47 = 280/2256
4 x 2 x 13/46 x 1/45 = 104/2070
896/2652 x 540/2450 x 280/2256 x 104/2070 = 1/2153
I'd like to know if I am wrong please.
Thank you for your answers
6 x 2 x 15/50 x 3/49 = 540/2450
5 x 2 x 14/48 x 2/47 = 280/2256
4 x 2 x 13/46 x 1/45 = 104/2070
896/2652 x 540/2450 x 280/2256 x 104/2070 = 1/2153
I'd like to know if I am wrong please.
Thank you for your answers
Romes
...Did you take into account that any of the 7 hands can make 4 blackjacks?
No, I simply 4 hands in a row getting blackjack, without replacement.Playing it correctly means you've already won.
ThatDonGuyThanks for this post from:
7 x 2 x 16/52 x 4/51 = 896/2652
6 x 2 x 15/50 x 3/49 = 540/2450
5 x 2 x 14/48 x 2/47 = 280/2256
4 x 2 x 13/46 x 1/45 = 104/2070
896/2652 x 540/2450 x 280/2256 x 104/2070 = 1/2153
I'd like to know if I am wrong please.
Thank you for your answers
You are counting every deal 24 times.
You appear to be saying, 'Any of the 7 players can have any of the four Aces, and for each one, any of the other 6 players can have any of the three remaining Aces,' but you are counting each hand where, for example, Player A has the Ace of Spades and Player B has the Ace of Hearts twice.
Any of the 7 players can have the Ace of Spades, but you should then be multiplying it by 1/52 instead of 4/52. Similarly with the Aces of Hearts, Clubs, and Diamonds.
gordonm888
Here are the 35 ways that 7 hands can be 4 Blackjacks and 3 non-blackjack hands.B = Black jack, X = Non-blackjack
BBBBXXX
BBBXBXX
BBBXXBX
BBBXXXB
BBXBBXX
BBXBXBX
BBXBXXB
BBXXBBX
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Blackjack Probability Math
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Blackjack Math And Probabilities
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This is probably not my most interesting post.
Blackjack Probability To Win
So many better men, a few of them friends, were dead. And a thousand thousand slimy things lived on, and so did I.