Ask the Wizard: Blackjack - Probability
How can I determine the odds of flat betting (no counting, no progressions , etc ) of being ahead in a negative game such as blackjack, w/o counting, with a 0.5% disadvantage after 45,000 or so hands? Is it even possible?
|Expected Values for 3-card 16 Vs. 10 in 8-deck game|
|Hand||EV Hit||EV Stand||Best |
The two right numbers in the bottom row show that the overall expected value for hitting is -0.540355 and for standing is -0.540293. So standing is the marginally better play. Following this rule will result in an extra unit once every 1117910 hands. It would take about 5 years playing blackjack 40 hours a week before this piece of advice saved the player one unit.
I play 6 deck blackjack in Tunica, MS. The dealer hits on soft 17. I wonder what the odds are of standing on 16 when the dealer’s upcard is 7. It seems only a 10 or face card can beat this and the odds would be in my favor if the dealer draws more than one card. Also, since most strategies are based on millions of calculations done on a computer, I wonder if those of us who will never play a million hands can rely on slight variations like this one. Is this a poor, fair or bad move to make?
— Richard S. from Memphis, USA
My friend and I are debating two blackjack issues that arose from his Caribbean Vacation. (1) What shift in odds does the dealer NOT drawing the second card have? House favor or player favor? (2) in your simulations, what impact does the number of players have on the accuracy of the odds?
— Beau from Toronto, Canada
In blackjack, what is the probability of a blackjack?
|Probability of Blackjack|
What is the probability that you play ten hands and never obtain a (two-card) 21? Assume the cards are reshuffled after each play?
— Matt from Radford, USA
What are the odds of getting 3 blackjacks in a row with 1 deck 4 players and one dealer.
— Joe P from Parma Heights, USA
In two handed blackjack using one deck, what is the probability of the dealer having a blackjack?
— Steve from Solva, United Kingdom
What are the odds of a dealer getting 3 blackjacks in a row on a single deck table with two players?
— J.A.S. from Las Vegas, USA
Dear Wizard, I was recently playing blackjack with somewhat of a card-shark who also happens to be my friend. We played casino rules, with one deck- and switched the deal after each time the deck expired. Later, while I was shuffling- I noticed two 9 of spades side by side. My friend obviously claimed he did not know about this, but it seems unlikely. My question is, if you were playing in a similar scenario and were to add one card to the deck, which card would be most advantageous if only you knew about it. Thank you for your time.
I’ve been playing blackjack for quite awhile using basic strategy, mostly betting an even unit each hand. Occasionally I will increase the bet because I "feel" like I am going to win the next one. I would think that just about all recreational players bet on feel once in a while at least. I was reading through some of your past Ask the Wizard columns and saw your calculation of the probability of a string of losses in the August 4, 2002 Column. You know those emotional thoughts that pop in head while gambling (well maybe not your head), "I’m due for a win!"
That column seemed to put the mathematics to that "feeling" a player can get. In that columns’ example of a player losing 8 consecutive hands of blackjack the odds were (.5251^8 or about 1 in 173). My question though is what does that really mean? Is it that when I sit down at the table, 1 out of my next 173 playing sessions I can expect to have an 8 hand losing streak? Or does it mean that on any given loss it is a 1 in 173 chance that it was the first of 8 losses coming my way?
I know, I know, its some sort of divine intervention betting system I am talking about and no betting system affects the house edge. I’m still curious though. Besides every once in awhile throwing down a bigger bet just adds to the excitement and for some reason it seems logical that if you have lost a string of hands you are "due" for a win.
— Steve from Phoenix, AZ
Dear wiz, I am a blackjack dealer here in Vegas and the other night dealing, I had 4 out of the 6 ace of spades in my hand. I had A-A-K-A-A-10, so good think is I busted, but quick calculations on the game, we figured getting 4 out of the six aces on one had is around 7mil to 1. Is this number a little high?
After performing my own infinite deck analysis for Blackjack with the same rules as yours (dealer stands all 17s, re-splitting allowed to 4 hands except Aces, which can only be split once, doubling after splitting, draw only one card to split Aces), I came across your site. In comparing expected values, I obtained the same numbers as you in all cases, except for pair splitting, which were slightly different. So I’m wondering how you went about your calculation of expected values for splitting?
- Take a 2 and two 8’s out of the shoe.
- Determine the probability that the player will not get a third eight on either hand.
- Go through all ranks, except 8, subtract that card from the deck, play out a hand with that card and an 8, determine the expected value, and multiply by 2. For each rank determine the probability of that rank, given that the probability of another 8 is zero. Take the dot product of the probability and expected value over each rank.
- Multiply this dot product by the probability from step 2.
- Determine the probability that the player will resplit to 3 hands.
- Take another 8 out of the deck.
- Repeat step 3 but multiply by 3 instead of 2.
- Multiply dot product from step 7 by probability in step 5.
- Determine the probability that the player will resplit to 4 hands.
- Take two more 8’s out of the shoe.
- Repeat step 3 but multiply by 4 instead of 2, and this time consider getting an 8 as a third card, corresponding to the situation where the player is forced to stop resplitting.
- Multiply dot product from step 11 by probability in step 9.
- Add values from steps 4, 8, and 12.
The hardest part of all this is step 3. I have a very ugly subroutine full of long formulas I determine using probability trees. It gets especially ugly when the dealer has a 10 or ace up.
Dear wiz, How do you calculate the probability of getting three sevens, three colored sevens, and three suited sevens in blackjack?
Good job and well done. The question: I notice from your May 5, 2003 Column that you actually CALCULATE your blackjack odds. I am a bit surprised that you were not using your computer to SIMULATE the results. Or is this a stupid question, i.e., the computer will take a million years to do the job?
I recently went to Vegas and had an incredible hand of blackjack... received an ace as first card, split, received another ace, split, received a third ace, split, and got one last ace... Then was dealt blackjack on all 4 hands! No lie! 2 of my friends were witnesses, as was the entire Luxor gaming gods...What are the odds on this? It was a 6 card deck shoe, I was sitting in #3 seat of a 4 person game. Assume a fresh shuffle?
I just witnessed a friend get four blackjacks in a row starting with the first hand of a newly shuffled single deck playing head to head against the dealer. I looked at the FAQ’s and saw the odds for getting one blackjack in single deck, but don’t know how to calculate them for getting four in a row off the top. Instead of a decimal probability, could you tell me the odds of this? It must be astronomical. Hope to hear from you.
First I wanted to tell you how much I look at and love your web site, and admire your math skills. I use 6 decks to deal blackjack, and added 3 jokers for reasons I won’t waste your time with but, what are the odds of dealing all 3 jokers to a player right in a row. Thank you very much.
Michael, a person asked you if they are not counting cards in blackjack, what difference does it make how many decks are being used. You stated the difference had mostly to do with the number of stiff hands possible, due to the fact that if a small card came out it was more likely a large card would follow and vice-a-versa. How could that be? Would it still not be a random event with the possibility of a small or large card coming out being equal, if you are not counting?
At a single deck game what is the probability all three players and the dealer get a blackjack the first round after a shuffle?
Player 1 0.048265
Player 2 0.036735
Player 3 0.024823
The product is 1 in 1,808,986.
Mr. Wizard, Great site. There is a lot of useful and interesting info. I’d like to see more of the mathematics and possible sources of simulations (source code, books, etc.) behind the games. Where would you suggest that a person interested in writing something similar to your "blackjack house edge calculator" go for more info? Thank you for your response.
What is the probability of a blackjack for n decks?
I am a blackjack dealer and last night I amazed my table on a single-deck blackjack game (the horrible 6 to 5). My hand consisted of an Ace up, Ace in the hole and then I drew the other 2 Aces and then a 7 for 21! What are the odds of this happening and I am especially interested in knowing the math. Thanks!
In blackjack, what is the probability of the dealer making a stopping hand (17-21) drawing eight cards? This happened to a friend of mine online and I think it's an extremely rare occurrence. How about seven cards? Thanks for the great site and keep up the awesome work!
Assuming a six-deck game, where the dealer stands on soft 17, and the player plays basic strategy here are the rounded results based on a 100-million hand simulation.
|Player Hand Probabilities|
|Dealer has only blackjack||1 in 22|
|Player doubles or splits||1 in 7.7|
|2 cards||1 in 2.3|
|3 cards||1 in 3.8|
|4 cards||1 in 10|
|5 cards||1 in 50|
|6 cards||1 in 400|
|7 cards||1 in 4,600|
|8 cards||1 in 79,000|
|9 cards||1 in 2,200,000|
|10 cards||1 in 100,000,000|
|Dealer Hand Probabilities|
|Player has only blackjack||1 in 22|
|2 cards||1 in 3.0|
|3 cards||1 in 2.4|
|4 cards||1 in 6.1|
|5 cards||1 in 31|
|6 cards||1 in 270|
|7 cards||1 in 3,700|
|8 cards||1 in 79,000|
|9 cards||1 in 2,200,000|
|10 cards||1 in 100,000,000|
If someone follows such a Martingale system in blackjack, what is the probability of being able to win $200 per day or lose the entire $5,000? Also, does increasing the amount available for total wagering increase the likelihood of winning the $200.
As I read your analysis of the Royal Match side bet in blackjack, am I correct that your odds are for the first hand of the shoe? If so, wouldn’t the real-world odds of an easy match be tilted a bit more toward the player? It seems to me that if the suits get unbalanced in any direction it would slightly lessen the house edge, and the suits will certainly fluctuate through the shoe.
— Frank from Michigan
I have been a dealer for 27 years and have seen a lot. One of my favorites was a guy who never looked at his cards playing blackjack....just tucked them. I thought he was nuts of course but some days he won and some days he lost. Just like most people. I tried this myself on a free gambling website and won 2 out of 3 times gambling 20 minute sessions. My question is this: How much worse off are you doing this than trying to play basic strategy? I really don’t think it matters much at all what you do in blackjack for the ’average’ player.
— Mark from Las Vegas
I lost a lot of money playing Cryptologic Blackjack today. While I don’t think anything is fixed, one aspect of my play seemed well outside the range of probability. Within 35 hands, the dealer showed a 6 seven times and won each time. This was verified through the logs. If the probability of a dealer bust is 56% with a six, my calculation suggests the odds of this independent event happening six consecutive times is 0.23%.
— Adam from Toronto
First off, my apologies if you consider this a basic math question. I’m a dealer at a Northern Ontario casino, and last night (for the dealer) drew a 12-card 17 (A-A-A-A-A-A-6-A-A-A-A-A). We use six decks. Neither my player or I had ever seen this before. What are the odds of this?
— Timothy Rowland from Orilila
I am a pit supervisor at a local casino and recently had a dealer deal two players two seven of clubs each and give himeself the last seven of clubs as his upcard on a five-deck shoe. What are the odds of five of the same card coming out of a five-deck shoe in order?
— Jesse from Scottsdale
According to standard BJ rules and perfect basic strategy, how many percent of my DOUBLED DOWN hands should I expect to win, push and lose?
— Cameron from Melbourne, Australia
I’ve played a lot of Blackjack over the years but have never struck anything like the situation I experienced on the weekend. Playing $25 a hand I lost 19 hands in a row with no pushes. One of the hands was a double down, so effectively I lost 20 x $25 bets in a row. I was playing strict Basic Strategy for New Zealand conditions (not counting, CSM in use). Have you ever heard of such a horror streak? I was ahead about $300 when the sky fell in but stuck to the strategy and eventually left the session $200 ahead and very relieved. My calculations estimate the probability of 19 straight losses as 1 chance in about 207,000; you may well correct me on this. I play to a betting progression system, purely for discipline/money management purposes which has me betting 1 unit after every loss. Had I done anything differently, I would have been cleaned out well before the 19 hands came up.
— Ken from Auckland, New Zealand
- Win 42.43%
- Lose 49.09%
- Draw 8.48%
So the probability of going exactly 19 losses in a row is 0.4909^19*(1-0.4909) = 1 in 1,459,921. By way of comparison, the probability of being dealt a royal flush in video poker is 1 in 649,740, or 2.25 times as likely. Avid video poker players have been known to receive several dealt royals, so if you play a lot of blackjack you’ll likely hit such a losing streak eventually.
I’m totally baffled! If in an 8-deck or continuous shuffle blackjack game there is no difference in the probabilities of a card appearing at any time, why have you posted Blackjack Appendix #18? If the probabilities say hit on 16 vs. 7 or higher, how can the probabilities change if you have 5 or 6 cards as opposed to 2 cards? 16 is 16 no matter how you construct it, right? I see the change if the deck is shrinking or in a game like Spanish 21 where there is a bonus for 21 with 5 or more cards, but why in an 8-deck game or continuous shuffle?
— Ernie from Toledo
In an eight-deck shoe there are 416 cards. That may seem like a lot, but 16 against a 10 is such a borderline hand that removal of just one card can making standing a better play. The rule is that for eight or fewer decks if your 16 is composed of three or more cards, and the dealer has a 10, then you should stand. In a two-card 16 the average points per card is 8, with a 3-card 16 the average is 5.33. With more small cards out of the deck in the 3-card hand the remaining deck becomes more large card rich, making hitting more dangerous, swaying the odds in favor of standing.
Hi Wizard. Thanks for maintaining this web site! I have a question about a blackjack rule that is applied in Dutch casinos: When being dealt a pair of sevens, a third seven will earn you 2:1 on your bet, regardless if you win the hand or not. However, this only applies when the sevens have NOT been split. I know that there are 6 dealer up cards in basic strategy that allow splitting sevens and 7 that do not, so the player should have an edge in this particular situation. But what are the odds of being dealt 3 sevens in blackjack in the first place? And if dealt 3 sevens, what are the odds they qualify for the 2:1 pay-out rule, based on a 4 to 6 decks, dealer stands on soft 17 basic strategy chart? Hope you can figure this one out for me. Keep up the good work!
— Stan from The Netherlands, Europe
I have a friend who starts complaining when his first card is a six, without waiting to see what his second card and the dealer’s up card are. I think he should wait because he could get a two, three, four, five, etc. (i.e., a decent second card) OR the dealer could show a two through six (a good card for the table). What do you think? How much worse are his odds of winning with a first card of six without knowing this second card or the dealer’s up card? Or is my friend just a whiner? Thank you for your time.
— Scott from Long Beach
Bally Gaming has a single-deck, multi-hand, blackjack game. The player plays seven hands against a single dealer hand. There is an interesting rule in that if the game runs out of cards, all unbusted player hands automatically win. What is the probability of running out of cards? Can have suggest any strategy changes to run out the deck?
— Michael L. from West Mifflin, PA
- Single deck.
- Dealer stands on soft 17.
- Winning blackjack pays even money.
- Player may double any first two cards.
- No double after split.
- Player may resplit to four hands, including aces.
- No draw to split aces.
- No surrender.
- Six-card Charlie (player unbusted six cards automatically wins).
- Cards shuffled after every hand.
- If game runs out of cards, all unbusted player hands automatically win.
The house edge using total-dependent basic strategy is 2.13%. I ran a 7-player simulation, using total-dependent basic strategy, and the average number of cards used per round was 21.65, with a standard deviation of 2.72. In almost 190 million rounds played, the most cards ever used was 42, which happened 7 times.
It is my educated opinion that even with computer perfect composition-dependent strategy the player would still realistically never see the last card. You could cut down the house edge much more using composition-dependent strategy, according to all the cards seen as you go along. However bucking 2.13% house edge to start with, you’ll never get anywhere near break-even, regardless of how hard you try.
Recently, the Tuscany casino ran a promotion where if you got 30 blackjacks in a 30-day period, you would win a $100 bonus. At first, the minimum bet was $5 to get your card stamped. However, I later heard the minimum for a stamp was raised to $15. I wrote a letter of complaint about it to the casino manager, stating in part:
I just wanted to express my disappointment in this change, if it is true. I never had a chance to take advantage of the promotion and doubt I will be able to now. The amount of time necessary to receive 30 blackjacks (I’m told about 8 hours of continuous play) seems unreasonable at $15/hand when the promotion still offers only $100.
Here is the reply I received:
In response to your e-mail on the blackjack blackout promotion, I’m not sure where you received your information on how long it takes to complete the blackout card. We’ve seen players complete the card in less than four hours. Also, you have thirty days in which to complete the card. I hope you understand this is not a task that is unreachable with that much time. I THANK YOU for your letter. It’s good to hear feedback from our customers. Hope you can give it a try and win some money!
What is the probability of getting 30 blackjacks in four hours?
The probability of filling the card in 4 hours, assuming 280 hands, is 1 in 30,000 playing one hand at a time. I suspect any player achieving the goal in four hours was playing at least two hands at a time.
This question was raised and discussed in the forum of my companion site Wizard of Vegas .
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