Blackjack Probability Worksheet

  • Appendices
  • Miscellaneous
  • External Links
  1. Blackjack Probability Theory
  2. Blackjack Probability Math
  3. Blackjack Math And Probabilities
  4. 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%.

  1. 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.
  2. 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.

Blackjack probability calculator

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
2345678910Ace
0-16-0.292784-0.252250-0.211063-0.167193-0.153699-0.475375-0.510518-0.543150-0.540430-0.666951
17-0.152975-0.117216-0.080573-0.0449410.011739-0.106809-0.381951-0.423154-0.419721-0.478033
180.1217420.1483000.1758540.1995610.2834440.3995540.105951-0.183163-0.178301-0.100199
190.3863050.4043630.4231790.4395120.4959770.6159760.5938540.2875970.0631180.277636
200.6399870.6502720.6610500.6703600.7039590.7732270.7918150.7583570.5545380.655470
210.8820070.8853000.8887670.8917540.9028370.9259260.9306050.9391760.9626240.922194

Hit

Player's Expected Return by HittingExpand

Player's
Hand
Dealer's Up Card
2345678910Ace
4-0.114913-0.082613-0.049367-0.0123800.011130-0.088279-0.159334-0.240666-0.289198-0.253077
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
7-0.109183-0.076583-0.043022-0.0072710.029185-0.068808-0.210605-0.285365-0.319055-0.310072
8-0.0217980.0080050.0387840.0708050.1149600.082207-0.059898-0.210186-0.249375-0.197029
90.0744460.1012650.1289810.1580320.1960190.1718680.098376-0.052178-0.152953-0.065681
100.1825000.2060880.2304700.2562590.2877950.2569090.1979540.1165300.0253090.081450
110.2383510.2603250.2830200.3073500.3336900.2921470.2299820.1582570.1194820.143001
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 120.0818360.1035070.1265960.1564820.1859540.1654730.0951150.000066-0.070002-0.020478
Soft 130.0466360.0741190.1024770.1333630.1616930.1223860.054057-0.037695-0.104851-0.057308
Soft 140.0223920.0508070.0800810.1118940.1391650.0795070.013277-0.075163-0.139467-0.093874
Soft 15-0.0001210.0291600.0592850.0919600.1182460.037028-0.027055-0.112189-0.173704-0.130027
Soft 16-0.0210250.0090590.0399750.0734490.098821-0.004890-0.066795-0.148644-0.207441-0.165637
Soft 17-0.0004910.0289750.0593260.0911890.1280520.053823-0.072915-0.149787-0.196867-0.179569
Soft 180.0629050.0902480.1185020.1476130.1907530.1706760.039677-0.100744-0.143808-0.092935
Soft 190.1239580.1493400.1755770.2029860.2397990.2206200.1522700.007893-0.088096-0.005743
Soft 200.1825000.2060880.2304700.2562590.2877950.2569090.1979540.1165300.0253090.081450
Soft 210.2383510.2603250.2830200.3073500.3336900.2921470.2299820.1582570.1194820.143001

Double

Player's Expected Return by DoublingExpand

Player's
Hand
Dealer's Up Card
2345678910Ace
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 8-0.204491-0.136216-0.0663720.0034560.087015-0.187730-0.451987-0.718501-0.746588-0.810746
Hard 90.0611190.1208160.1819490.2430570.3170550.104250-0.026442-0.300996-0.466707-0.432911
Hard 100.3589390.4093210.4609400.5125170.5755900.3924120.2866360.144328-0.008659-0.014042
Hard 110.4706410.5177950.5660410.6146990.6673800.4628890.3506930.2277830.1796890.109061
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 12-0.071570-0.0072280.0584270.1259540.179748-0.183866-0.314441-0.456367-0.514028-0.624391
Soft 13-0.071570-0.0072280.0584270.1259540.179748-0.183866-0.314441-0.456367-0.514028-0.624391
Soft 14-0.071570-0.0072280.0584270.1259540.179748-0.183866-0.314441-0.456367-0.514028-0.624391
Soft 15-0.071570-0.0072280.0584270.1259540.179748-0.183866-0.314441-0.456367-0.514028-0.624391
Soft 16-0.071570-0.0072280.0584270.1259540.179748-0.183866-0.314441-0.456367-0.514028-0.624391
Soft 17-0.0070430.0550950.1186530.1823780.256104-0.013758-0.255102-0.400984-0.458316-0.537198
Soft 180.1197500.1776410.2370040.2952250.3815060.219948-0.029917-0.290219-0.346892-0.362813
Soft 190.2418550.2958240.3511540.4059720.4795990.3198350.195269-0.072946-0.235468-0.188428
Soft 200.3589390.4093210.4609400.5125170.5755900.3924120.2866360.144328-0.008659-0.014042
Soft 210.4706410.5177950.5660410.6146990.6673800.4628890.3506930.2277830.1796890.109061

Split

Player's Expected Return by SplittingExpand

Player's
Hand
Dealer's Up Card
2345678910Ace
2,2-0.084336-0.0156500.0590880.1516650.2268900.006743-0.176693-0.386883-0.507175-0.433570
3,3-0.137710-0.0562730.0299320.1262840.201318-0.053043-0.231843-0.436607-0.553507-0.482405
4,4-0.192325-0.108712-0.0203950.0819130.151377-0.166452-0.326068-0.511152-0.625044-0.560206
5,5-0.290154-0.208718-0.119335-0.0192310.045404-0.293928-0.454237-0.634113-0.729969-0.668811
6,6-0.212560-0.119715-0.0213200.0809120.153668-0.264427-0.425122-0.610576-0.716103-0.653362
7,7-0.131478-0.0437330.0492550.1466780.247385-0.050148-0.391981-0.577584-0.657268-0.651641
8,80.0738520.1461870.2208490.2974750.4093290.321042-0.022736-0.387228-0.480686-0.372535
9,90.1956250.2585480.3234740.3919870.4713390.3648370.234447-0.078010-0.317336-0.136810
10,100.1347740.2128360.2934030.3803670.4681170.2966330.064443-0.206733-0.371278-0.249494
A,A0.4706410.5177950.5660410.6146990.6673800.4628890.3506930.2277830.1796890.109061

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.
Hunterhill
I don't know the odds, but this has to be one of the worst bonuses I've heard of.
Romes
Thanks 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.
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?
ThatDonGuy
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.

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
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.
ThatDonGuy
Thanks 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
BBXXBXB
BBXXXBB
BXBBBXX
BXBBXBX
BXBBXXB
BXBXBBX
BXBXBXB
BXBXXBB
BXXBBBX
BXXBBXB
BXXBXBB
BXXXBBB
XBBBBXX
XBBBXBX
XBBBXXB
XBBXBBX
XBBXBXB
XBBXXBB
XBXBBBX
XBXBBXB
XBXBXBB

Blackjack Probability Math


XBXXBBB
XXBBBBX

Blackjack Math And Probabilities

XXBBBXB
XXBBXBB
XXBXBBB
XXXBBBB
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.