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1. Blackjack Probability To Win
2. Blackjack Probability Calculator
3. Blackjack Probability Theory
4. Blackjack Probability Worksheet
5. Blackjack Math And Probabilities

When the dealer stands on a soft 17, the dealer will bust about 29.1% of the time. When the dealer hits on a soft 17, the dealer will bust about 29.6% of the time. According to my blackjack appendix 4, the probability of a net win is 42.42%. However, if we skip ties, the probability is 46.36%. In a freshly shuffled deck (standard 52 cards), what is the probability that neither of you are dealt a blackjack. Blackjack being 2 cards adding to 21 i.e. A c e + 10, J, Q, o r K (or vice versa as order does not matter). The farthest I've really come is that the odds of the first player getting dealt a blackjack. How to calculate the probability of an event. The first concept to understand is that probability is. Blackjack In the game of blackjack played with one deck, a player is initially dealt 2 different cards from the 52 different cards in the deck. A winning “blackjack” hand is won by getting 1 of the 4 aces and 1 of 16 other cards worth 10 points. The two cards can be in any order. Find the probability of being dealt a blackjack hand.

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I would divide this 'last card analysis' into 4 scenarios:
1. The facedown card of the dealer is the last card because all players had initial hands of 17-21 and dealer has a 17 -20.
2. Dealer is last to draw a card because 1 or more players have not busted and the dealers original 2 card hand was 16 or less.
3. A player is the last to draw a card because player had a hand that required a Hit and:
a) dealer's first two cards are 17 -20
b) OR, dealers first two cards were <17 but all players HIT and busted.
4. Dealer gets a blackjack, peeks, and turns his cards over ending the hand. (Last card is Ace or Ten with equal probability.)
Under scenario 1, there is a bias: the last card must be a 7, 8, 9, T or A.
In scenarios 2 and 3 there is also a bias towards high cards because of the possibility of multicard draws. When the dealer or player is hitting a hand that 7-11, than a high card may possibly end the drawing process immediately, while a low card will require the dealer/player to hit again. Even if say, dealer or player has a hard 15 and hits, then a 2-T will end the hand ,whereas an Ace will require a re-draw - so in that scenario a 2-T has a higher probability of being the last card than an ace would.
I will note that the probability of scenarios 1 - 3 are strongly dependent upon the number of players at the table.
Edited: inserted scenario 4, Dealer blackjack.I imagine someone will just simulate this problem and get the bottom-line answer, but I find that working through this kind of problem is educational.
Let's assume 1 player, infinite deck and H17, DOA2
For scenario 1: The facedown card of the dealer is the last card because all players (the only player) stood on initial hands of 17-21 and dealer has a H17-20.
This scenario will occur with a probability of 0.067295.
The last card drawn in this scenario is
Ace: 0.005042_____7.4922%
Ten: 0.037289____ 55.411%
9: 0.011834_____17.5858%
8: 0.009839_____14.6202%
7: 0.003291______4.8907%
This was tricky to calculate, because of the need to factor in that the following hands are not in the scenario: 99 vs 8,9 (Split) and A8 vs 9,T,A (HIT)First, the result was for Hit 17.
Nanumula, welcome to the forums. You posted an interesting question and some of us have spent a little time looking at aspects of it. But also realize that we are thinking about the question you posted out of:
1. Good will
2. Curiosity.
It is a bit unreasonable to suppose that forum members are going to do complex calculations/simulations as a function of number of decks and rule variations, etc. simply because a new member has posted and asked.
It would help if you would tell us why you are asking. Maybe tell us a little background info about yourself.

First, the result was for Hit 17.
Nanumula, welcome to the forums. You posted an interesting question and some of us have spent a little time looking at aspects of it. But also realize that we are thinking about the question you posted out of:
1. Good will
2. Curiosity.
It is a bit unreasonable to suppose that forum members are going to do complex calculations/simulations as a function of number of decks and rule variations, etc. simply because a new member has posted and asked.
It would help if you would tell us why you are asking. Maybe tell us a little background info about yourself.

If I were a betting man I would wager the OP has an idea for a BJ bonus bet.

If I were a betting man I would wager the OP has an idea for a BJ bonus bet.

Lol, that could also be a good idea for the future. Although, I can't implement it since I don't work for the casino. It would be fun though to design new sidebets with huge house advantages and just rake in the dough from the casino side haha.

...interesting question...we are thinking about the question you posted out of:
1. Good will
2. Curiosity. ...

I agree it is an interesting puzzle in terms of working out how one might attack the calculation for a single player. However I think it gets too complicated with multiple players. So thanks for exercising the brain cells!
If I was continuing past that, I would write code to work out all the reasonable permutations (initially ignoring the effect of re-splits) and list the last card used assuming there was only one player (having more players is too much work at this stage).

Blackjack Probability To Win

If it was a phD then I should run simulations to confirm the calculations of one player and then add more players to see whether that affects anything (in practice it would mean the dealer plays out their cards more often).
Given the last part there might be a thought that one works out, given a number of players, how often the dealer does or doesn't play out their hand given any of their possible upcards (and not having BJ). For instance if the dealer has a 4, 5 or 6 faceup (except 6A s17) then the only thing to consider is all players get a BJ, otherwise whatever happens they don't bust and the dealer will act last.
Using UK rules (no peek, DDAS, s17, 6 decks) I got the following (you can see from the different counts for Kings, Queens etc how much of an estimate this gives).
Note I merely added code to set lastcarddrawn in the getacard routine and when the hand was over tallying the values of lastcarddrawn before moving onto the next hand; so I haven't checked the results in any detail.

...new sidebet...[on the last card drawn]

You could design a sidebet based on the last card the dealer drew (that is easy to work out assuming you always play the dealer's hand to a finish, similar to the '22' bet in freebet). The problem if you includes the player's cards is it might affect their decisions, for instance the last player to act has a clear advantage as they could stand on their 2-card hand or deliberately take another card.

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Blackjack Probability and Blackjack Odds

To fully understand the game of blackjack, you must understand and master blackjack odds. It's crucial to know how the casino gains its edge and how it helps them win. It's also important to understand blackjack odds such as the odds of hitting a 10 or the odds of being dealt a blackjack. You can read over these blackjack odds charts to help understand the logistics behind blackjack.

Casino and Dealer Advantage in Blackjack

In most casino games of blackjack, the house advantage (the dealer advantage) is ~ 8%. The house gets this advantage by the dealer being the last player to act. By acting last, all other players have already made their decisions and could quite possibly bust before the dealer has his turn.

By using correct blackjack basic strategy, you can turn the casino edge in blackjack from 7%-8% down to 0.5%. If you correctly use advanced card counting techniques, you can often change blackjack odds and give yourself the advantage over the casino. Manipulating the house odds to your favor is the reason most casinos don't allow card counters to play blackjack.

Probability of Busting on a Hit

It's very important to know the probability of your hand busting when you are holding any total in the game of blackjack. The following odds chart shows the blackjack odds of busting, depending on your current hand value:

Hand Value	% Bust If You Hit
21	100%
20	92%
19	85%
18	77%
17	69%
16	62%
15	58%
14	56%
13	39%
12	31%
11 or Less	0%

Two-Card Count Frequencies

This interesting blackjack odds chart is the two card count frequency chart. This chart shows the percentage chance that you will be dealt a hand in each given value range. The most important frequencey to note is the chance of being dealt a natural blackjack (natural 21 value. The odds of being dealt a natural blackjack are merely 4.8%. Following this chart you will see that the most common two card hand, at 38.7%, is a hand totaling 1-16, which is considered a decision hand.

Two Card Count	% Frequency
Natural 21	4.8%
Hard Standing (17-20)	30.0 %
Decision Hands (1-16)	38.7%
No Bust	26.5%
TOTAL	100.0%

Blackjack Probability Calculator

Dealer Final Hand Probabilities

This blackjack odds chart shows the dealer final hand probability. These are the percentages that the dealer will end up with a hand totaling each corresponding value (up to 16). Read over this chart to understand the odds that the dealer has to make his final hand.

Dealer Final Hand Value	%	Cumulative % Total
Natural 21	4.82%	4.83%
21 (3 or More Cards)	7.36%	12.19%
20	17.58%	29.77%
19	13.48%	43.25%
18	13.81%	57.06%
17	14.58%	71.64%
16	28.36%	100.00%

Player Advantage vs. Dealer Up Card

The first two columns in this odds chart explain the dealer's chance of busting, depending on the up card that he is showing. You should note that the dealer has the highest chance of busting when he is showing a 5. The third column in this chart shows the player advantage of using basic strategy, compared to each up card the dealer is showing. You can see that the player has the highest advantage of 23.9%, when the dealer is showing a 5. When the dealer is showing any card that is 9 value or higher, the player is in the negative advantage range.

Dealer Up Card	Dealer Bust %	Player Advantage % with Basic Strategy
2	35.30%	9.8%
3	37.56%	13.4%
4	40.28%	18.0%
5	42.89%	23.2%
6	42.08%	23.9%
7	25.99%	14.3%
8	23.86%	5.4%
9	23.34%	-4.3%
J,Q,K	21.43%	-16.9%
A	11.65%	-16.0%

Effects of Removing Cards from a Deck

Blackjack Probability Theory

When looking at the odds of removing certain cards from a 52-card deck, some cards have a much greater effect on blackjack odds. To create the strongest card counting system ever invented, you would have to incorporate all of these slight and subtle differences into the numbers to be a completely accurate system.

Removing every 5 from a deck cards would make the largest impact of improving your blackjack odds, as a player. On the other hand, removing every Ace from a deck of cards would make the largest impact on improving the odds for the casino.

Card	% Effect of Removal
2	0.40%
3	0.43%
4	0.52%
5	0.67%
6	0.45%
7	0.30%
8	0.01%
9	-0.15%
10	-0.51%
A	-0.59%

Blackjack Probability Worksheet

See also:

Blackjack Math And Probabilities