# What is the probability of being dealt a Blackjack

## What is the probability of being dealt a Blackjack?

When the deck is full, this probability is 4/52 (the probability of getting one of 4 aces) times 16/51 (the probability of getting one of 16 kings, queens, jacks, or tens) times 2 (the number of orders in which you could get these cards: ace first, or ace second). This comes out to 32/663, or about 4.83%. Of course, this probability changes as the game progresses: it decreases when any of the tens, jacks, queens, kings, or aces get discarded, but increases when other cards get discarded. This change is unpredictable, but its expected value is 0; this is a complicated concept to explain, but it means that on average, the probability will go up as much as it goes down. Also, the probability is still 32/663 at any point in the game if you have no information whatsoever about what cards came up before: if you forgot every card you saw, or if you just joined the game.

## Dealer and Player Odds in Blackjack

Odds are everywhere you look. Seriously. Odds are in a casino, your workplace, your day-to-day live and habits, your marriage and where in the world you live.

Dont believe me? Take a look for yourself:

• Odds of having twins (21st century) 3 in 100 or 3%
• Odds of getting a divorce 40 to 50%
• Odds of being stuck by lightning 1 in 5,000 or 1 in 700,000 in any given year
• Odds of being audited by the IRS .4% if you make less than \$200,000 / year
• Odds of being dealt a blackjack About 4.83%

Odds are just the likelihood that something will happen. As a blackjack player you deal with this all the time.

Lets look at a couple real examples to show you what I mean.

### Odds of Busting

Here are the odds of you busting your hand, depending on what you were dealt:

• 21 100%
• 20 92%
• 19 85%
• 18 77%
• 17 69%
• 16 62%
• 15 58%
• 14 56%
• 13 39%
• 12 31%
• 11 0%

### Odds of Being Dealt Specific Hands

Here are the probabilities for being dealt a specific hand:

• Blackjack 4.8%
• Standing Hand (17-20) 30%
• Decision Hand (1-16) 38.7%
• No Bust 26.5%

### Final Hand Odds for Dealers

Here are the odds for the final hands that the dealer will make:

• Natural 21 4.82%
• 21 (3+ cards) 7.36%
• 20 17.58%
• 19 13.48%
• 18 13.81%
• 17 14.58%
• 16 28.36%

### Dealer vs. Player Odds

Finally, here are the odds of the dealer busting based on their up card:

• 2 35.30%
• 3 37.56%
• 4 40.28%
• 5 42.89%
• 6 42.08%
• 7 25.99%
• 8 23.86%
• 9 23.34%
• J,Q,K 21.43%
• A 11.65%

Of these examples, this is the most useful. Notice what hands the dealer is most likely to bust with. The dealer will most often bust with 4, 5 or 6, followed by 2 and 3.

### How to Turn the Odds in Your Favor

The odds above are static. Theres nothing you can do to change them. However, you can find ways to improve your odds so that you lose fewer hands and less money. And the less money you lose, the more you can keep to play more blackjack.

Heres what you can do to improve your odds in blackjack:

• Use basic blackjack strategy. Without basic strategy youre playing at an 8% disadvantage. With strategy, however, the odds increase significantly. Youll be playing at only .5 to 1.5% disadvantage.
• Find the best games. The rules make all the difference. For example, if you play a 6:5 blackjack game youre adding a 1.39% disadvantage. If the dealer hits soft 17 thats another .18%. However, its possible to find games where the player is paid 3:2 for blackjacks and the dealer stands on soft 17. So find those games. Also be careful about which blackjack variation you play. The best games to play are blackjack, Spanish 21, Vegas
Strip BJ and Blackjack Switch.
• Avoid side bets. Side bets look exciting, because for a nominal amount you could possibly win a huge jackpot — worth hundreds or even thousands of dollars. The problem is that the odds against winning can be anywhere from 2-40%.
• Avoid wives tales. There are a lot of bad strategies like never busting (never hit a 12+), mimicking the dealer and assume that the dealer has a 10 in the whole (with ace up). The problem with using any of these strategies is that they increase the house edge from 3 to 10 percent.

You can do other things, too, like count cards or read books (usually a mix of basic strategy, card counting and general how-tos for casino blackjack). However, youll improve your odds at winning at blackjack just by following my suggestions above.

### Understanding the Long Run Sample Size and Variance

I wanted to finish up this article with a brief explanation of odds, and how they work over the long run. You see, I think a lot of people will see the numbers above and get confused when they don t match their own stats. In other words, someone might go to the casino play 500 hands of blackjack, and wonder why they didnt get 24 natural blackjacks, or the other way around, why they got 42.

The thing is, odds and statistics are all about the long run. Long run usually meaning sample size, or the total number of hands (or games) played.

What that means is that over a significant sample size, hundreds of thousands or even millions of hands, the number of times youll receive a blackjack is about 4.82%. The more hands you play the truer this will become.

The reason why odds dont match up in smaller sessions, say over 500 hands, is because of variance. Theres a technical term and definition for variance, but Ill just give you my version; variance is the ups and downs you experience on your way to the long term (expected) results.

Mike Caro, a poker player and author, puts it this way:

A measure of the spread of statistical distribution about its mean or centre.

That means in a short time frame, its possible to experience more drastic odds. You might win or lose more than youre supposed to. It also explains why people can go into a casino, not use basic strategy and win 3x as much as what they walked in with. The cards ran in their favor — they experienced a positive streak of variance.

So thats the gist of it. So the next time you walk into the casino and have a wild swing one way or another, you know that thats not normal, and that in the long run youll be closer to break-even so long as you stick to basic strategy — the plan with the best odds.

## Probability of being dealt a blackjack

We first present the probabilities attached to card dealing and initial predictions. In making this calculus, circumstantial information such as fraudulent dealing is not taken into account (as in all situations corresponding to card games). All probabilities are calculated for cases using one or two decks of cards. Let us look at the probabilities for a favorable initial hand (the first two cards dealt) to be achieved. The total number of possible combinations for each of the two cards is C(52, 2) = 1326, for the 1-deck game and C(104, 2)=5356 for the 2-deck game.

Probability of obtaining a natural blackjack is P = 8/663 = 1.20663% in the case of a 1-deck game and P = 16/1339 = 1.19492% in the case of a 2-deck game.

Probability of obtaining a blackjack from the first two cards is P = 32/663 = 4.82654% in the case of a 1-deck game and P = 64/1339= 4.77968% in the case of a 2-deck game.

Similarly, we can calculate the following probabilities:

Probability of obtaining 20 points from the first two cards is P = 68/663 = 10.25641% in the case of a 1-deck game and P = 140/1339 = 10.45556% in the case of a 2-deck game.

Probability of obtaining 19 points from the first two cards is P = 40/663 = 6.03318% in the case of a 1-deck game and P = 80/1339 = 5.97460% in the case of a 2-deck game.

Probability of obtaining 18 points from the first two cards is P = 43/663 = 6.48567% in the case of a 1-deck game and P = 87/1339 = 6.4973% in the case of a 2-deck game.

Probability of getting 17 points from the first two cards is P = 16/221 = 7.23981% in the case of a 1-deck game and P = 96/1339 = 7.16952% in the case of a 2-deck game.

A good initial hand (which you can stay with) could be a blackjack or a hand of 20, 19 or 18 points. The probability of obtaining such a hand is calculated by totaling the corresponding probabilities calculated above: P = 32/663 + 68/663 + 40/663 + 43/663 = 183/663, in the case of a 1-deck game and P = 64/1339 + 140/1339 + 80/1339 + 87/1339 = 371/1339, in the case of a 2-deck game.

Probability of obtaining a good initial hand is P = 183/663 = 27.60180% in the case of a 1-deck game and P = 371/1339 = 27.70724% in the case of a 2-deck game.

The probabilities of events predicted during the game are calculated on the basis of the played cards (the cards showing) from a certain moment. This requires counting certain favorable cards showing for the dealer and for the other players, as well as in your own hand. Any blackjack strategy is based on counting the cards played. Unlike a baccarat game, where a maximum of three cards are played for each player, at blackjack many cards could be played at a certain moment, especially when many players are at the table. T hus, both following and memorizing certain cards require some ability and prior training on the player?s part. Card counting techniques cannot however be applied in online blackjack .

The formula of probability for obtaining a certain favorable value is similar to that for baccarat and depends on the number of decks of cards used. If we denote by x a favorable value, by nx the number of cards showing with the value x (from your hand, the hands of the other players and the face up card in the dealer?s hand) and by nv the total number of cards showing, then the probability of the next card from the deck (the one you receive if you ask for an additional card) having the value x is:

This formula holds for the case of a 1-deck game. In the case of a 2-deck game, the probability is:

Generally speaking, if playing with m decks, the probability of obtaining a card with the value x is:

Example of application of the formula: Assume play with one deck, you are the only player at table, you hold Q, 2, 4, A (total value 17) and the face up card of the dealer is a 4. Let us calculate the probability of achieving 21 points (receiving a 4).

We have nx = 2, nv = 5, so:

For the probability of achieving 20 points (receiving a 3), we have nx = 0, nv = 5, so:

For the probability of achieving 19 points (receiving a 2), we have nx = 1, nv = 5, so:

If we want to calculate the probability of achieving 19, 20 or 21 points, all we must do is total the three probabilities just calculated. We obtain P = 9/47 = 19.14893%.

Unlike in baccarat, where fewer cards are played, the number of players is constant (two), and the number of gaming situations is very limited, in blackjack, the number of possible playing configurations is in the thousands and, as a practical matter, cannot be entirely covered by tables of values.

## Blackjack Probability and Statistics: A Complete Guide

Probability is measuring the possibility of occurring a particular event. Probabilities are usually expressed as a percentage. However, fractions and odds can also denote it.

Blackjack is a purely mathematical game and is all about probability and odds. Blackjack Hall of Famer Edward Thorp was among the earlier players who realised this phenomenon in the 50s and created a statistical simulation to increase his chances of winning.

In the short run, it is almost impossible to predict a result, but in the long term, outcomes can match with prediction. So, today, let us explore the ways of predicting Blackjack probability. Stay with us to know more on this subject.

### Understanding Blackjack Probability

Probability is the division of mathematics that deals with the occurrence of a specific event. It is always a digit between 0 and 1, where 0 denotes, the event will never happen, and 1 indicates, the event is bound to happen. Probability can be expressed in odds. For example, an event with a 50% probability has 1 to 1 odd or even odd.

Calculating probability in odds is useful in Blackjack, especially when you want to know whether you have the edge over the house or not. For example, you, having a 1 to 1 odds in a \$1 bet, means you have an expected value of zero so, you won t gain or lose any money over a long time.

Every casino game has a negative expected value. That is how casinos stay in profit. You cannot beat the house in the long run. However, you can study the previous rounds and make a positive expected value for yourself in one particular bet. That s how card counters operate and make a fortune in Blackjack.

### Independent and Dependent Trial

Before moving into all the calculations, let us give a brief introduction about two types of contrasting events, that can be seen happening in a typical casino game. An independent event is one that does not affect the upcoming events. Like the dice toss in Craps, the probability of throwing a 2 in a six-sided dice will always be ?. It doesn t change with any previous outcome.

On the other hand, in the case of dependent trials, probability changes with each round of bets. Blackjack is the prime example of a dependent trial. With each card dealt, the probability of getting any of the remaining card increases.

### The Probability of Getting a Blackjack

Naturals are the most potent combination you can get. With a natural in your disposal, not only it ensures your win in most of the cases, but you ll get 1.5 times your original bet. This is why understanding the probability of getting a blackjack is essential.

If you know the number of card decks in play, you can effortlessly determine the possibility of receiving a natural. For this purpose, you have to multiply the probability of getting an ace with the probability of pulling a ten-valued card, i.e., 10, J, Q, K. You also have to double up the result to get the actual probability. There are two probable combinations of the card in a Blackjack hand, for example, A-K and K-A.

4/52 is the probability of getting an ace while the possibility of pulling a ten-valued card from the rest of the deck in 16/51. Therefore, the probability of getting a natural in the first hand itself is

• 2*P(ace)*P(10-valued cards)
• 2*4/52*16/51 = 0.0482 or 4.82%

### Difference Between the Probability of Blackjack and Other Casino Games

The main difference between Blackjack and other games is players can get a strategic edge in Blackjack. With each card being dealt with, the composition of the remaining deck changes. If someone with an identical memory can memorise the cards which have already been dealt with, then he/she can have an idea about the cards that are on the way.

For example, if in the first couple of hands, all four aces are out, then your probability of getting a natural is 0. In the same way, if you are getting lower value cards at the start of the game, then you can assume higher value cards are coming and can plan your moves accordingly.

However, when you play Roulette or Craps, the odds won t change with every outcome. That s because Roulette wheels will not eliminate a number that has already come previously. You will always have an equal probability for all 38 numbers are coming up.

### Final Thoughts

Blackjack probability is an amusing yet vast subject with no end of topics to discuss. In this article, we could barely scratch the upper surface. This is just an aerial view of the subject. If your ultimate goal is to get success as a card counter, you can read many exciting journals on this topic.

However, for most people, card counting can be a daunting task. It may sound easy in theory, but in reality, only a fraction of our popularity can remember cards in a fast-paced casino game. Casino s strict monitoring policy to prevent card counters has made the task even more challenging nowadays.

Apart from being a card counter, learning the probability associated with Blackjack can help you understand the game better than just playing with blank eyes. You will have a clear concept of how a NZ online casino makes money and on which situation you can beat them. So keep expanding your knowledge on the subject.

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