Calculate Blackjack Probability, Odds: Natural 21, Insurance, Double-Down Hands, Pairs

2) How about multiple decks of cards? The calculations are not exactly linear because of the combination formula. For example, 2 decks, (104 cards):

~ the 2-deck case:

C(52, 2) = 1326

C(104, 2) = 5356 (4.04 times larger than total combinations for one deck.)

8 (Aces) * 32 (Tens) = 256

Odds of BJ for 2 decks = 256 / 5356 = 4.78% (a little lower than the one-deck case of 4.83%).

~ the 8-deck case, 416 total cards:

C(52, 2) = 1326

C(416, 2) = 86320 (65.1 times larger than total combinations for one deck.)

32 (Aces) * 128 (Tens) = 4096

Odds of BJ for 8 decks = 4096 / 86320 = 4.75% (a little lower than the two-deck situation and even lower than the one-deck case of 4.83%).

There are NO significant differences regarding the number of decks. If we round the figures, the general odds to get a natural blackjack can be expressed as 4.8%.

The advantage to the blackjack player after cards were played: Not nearly as significant as the one-deck situation.

3) The position at the table is inconsequential for the blackjack player. Only heads-up and one deck of cards make a difference as far the improved odds for a natural are concerned.

• Axiomatic one, let's cover all the bases, as it were. The original question was, exactly, as this: "Dealing yourself a blackjack (Ace AND Face-card or Ten) from a single deck". The calculations above are accurate for this unique situation: ONE player dealing cards to himself/herself. The odds of getting a natural blackjack are, undoubtedly, 1 in 21 hands (a hand consisting of exactly 2 cards).
• Such a case is non-existent in real-life gambling, however. There are at least TWO participants in a blackjack game: Dealer and one player. Is the probability for a natural blackjack the same – regardless of number of participants? NOT! The 21 hands (as in probability p = 1 / 21) are equally distributed among multiple game agents (or elements in probability theory). Mathematics — and software — to the rescue! We apply the formula known as exactly M successes in N trials. The best software for the task is known as SuperFormula (also component of the integrated Scientia software package).
  

• Undoubtedly, your chance to get a natural BJ is higher when playing heads-up against the dealer. The degree of certainty DC decreases with an increase in the number of players at the blackjack table. I did a few calculations: Heads-up (2 elements), 4 players and dealer (5 elements), 7 players and dealer (8 elements).
  • The degree of certainty DC for 2 elements (one player and dealer), one success in 2 trials (2-card hands) is 9.1%; divided by 2 elements: the chance of a natural is 9.1% / 2 = 4.6% = the closest to the "Dealing yourself a blackjack (Ace AND Face-card or Ten) from a single deck" situation.
  • The chance for 5 elements (4 players and dealer), one success in 5 trials (2-card hands) is 19.6%; distributed among 5 elements, the degree of certainty DC for a blackjack natural is 19.6% / 5 = 3.9%.
  • The probability for 8 elements (7 players and dealer), one success in 8 trials (2-card hands) is 27.1%; equally distributed among 8 elements, the degree of certainty DC of a blackjack natural is 27.1% / 8 = 3.4%.
• That's mathematics and nobody can manufacture extra BJ natural 21 hands... not even the staunchest and thickest card-counting system vendors! The PI... er, pie is small to begin with; the slices get smaller with more mouths at the table. Ever wondered why the casinos only offer alcohol for free — but no pizza?

1.2. Probability, Odds for a Blackjack Playing through a Deck of Cards

The probabilities in the first chapter were calculated for one trial. That is, the odds to get a blackjack in the first two cards. But what are the probabilities to get a natural 21 dealing an entire deck?

1.2.A. Dealing 2-card hands until the deck is dealt entirely

• The probability of at least one success in 26 trials for an event of individual probability p=0.0483 is 72.39%.

1.2.B. Dealing 2-card hands in heads-up play until the deck is dealt entirely

• The probability of at least one success in 9 trials for an event of individual probability p=0.0483 is 35.95%.

You, the player, can expect one blackjack every 3 decks in heads-up play.

2. House Edge on the Insurance Bet at Blackjack

We can devise precise mathematical formulas based on the Tens remaining in the deck. We know for sure that the casino pays 2 to 1 for a successful insurance (i.e. the dealer does have Ten as her hole card).

We start with the most easily manageable case: One deck of cards, one player, the very beginning of the game. There is a total of 16 Teens in the deck (10, J, Q, K). The dealer has dealt 2 cards to the player and one card to herself that we can see exactly — the face card being an Ace. Therefore, 52 – 3 = 49 cards remaining in the deck. There are 3 possible situations, axiomatic one:

• 1) The player has 2 non-ten cards; there are 16 Teens in the deck = the favorable situations to the player if taking insurance. There are 49 – 16 = 33 unfavorable cards to insurance. However, the 16 favorable cards amount to 32, as the insurance pays 2 to 1. The balance is 33 – 32 = +1 unfavorable situation to the player but favorable to the casino (the + sign indicates a casino edge). In this case, there is a house advantage of 1/49 = 2%.
• 2) The player has 1 Ten and 1 non-ten card; there are 15 Teens remaining in the deck = the favorable situations to the player if taking insurance. There are 49 – 15 = 34 unfavorable cards to insurance. However, the 15 favorable cards amount to 30, as the insurance pays 2 to 1. The balance is 34 – 30 = +4 unfavorable situations to the player but favorable to the casino. In this case, there is a house advantage of 4/49 = 8%.
  • This can be also the case of insuring one's blackjack natural: an 8% disadvantage for the player.
  • This figure of 8% represents the average house edge regarding the insurance bet. I did calculations for various situations — number of decks and number of players.
• 3) The player has 2 Ten-count cards; there are 14 Teens in the deck = the favorable situations to the player if taking insurance. There are 49 – 14 = 35 unfavorable cards to insurance. However, the 14 favorable cards amount to 28, as the insurance pays 2 to 1. The balance is 35 – 28 = +7 unfavorable situations to the player but favorable to the casino. In this case, there is a house advantage of 7/49 = 14%. This is the worst-case scenario: The player should never — ever — even think about insurance with that strong hand of 2 Tens!

Believe it or not, the insurance can be a really sweet deal if there are multiple players at the blackjack table! Let's say, 5 players, the very beginning of the game. There is a total of 16 Teens in the deck (10, J, Q, K). The dealer has dealt 10 cards to the players and one card to herself that we can see exactly — the face card being an Ace. Therefore, 52 – (10 + 1) = 41 cards remaining in the deck. There are many more possible situations, some very different from the previous scenario:

• 1) The players have 10 non-ten cards; there are still 16 Tens in the deck = the favorable situations to the player if taking insurance. There are 41 – 16 = 25 unfavorable cards to insurance. However, the 16 favorable cards amount to 32, as the insurance pays 2 to 1. The balance is 25 – 32 = –7 favorable situation to the player but unfavorable to the casino (the – sign indicates a player advantage now). In this case, there is a house advantage of 7/41 = –17%. The Player has a whopping 17% advantage if taking insurance in a case like this one!
• 2) The players have 10 Ten-count cards; there are 6 Teens in the deck = the favorable situations to the player if taking insurance. There are 41 – 6 = 35 unfavorable cards to insurance. However, the 6 favorable cards amount to 12, as the insurance pays 2 to 1. The balance is 35 – 12 = +23 unfavorable situations to the player but favorable to the casino. In this case, there is a house advantage of 23/41 = 56%. This is the worst-case scenario: None of the players should ever even think about insurance with those strong hands of 2 Tens per capita!
• 3) Applying the wise aurea mediocritas adagio, there should be an average of 3 or 4 Teens coming out in 11 cards; thus, 12 or 13 Tens remaining in the deck. There are 41 – 13 = 28 unfavorable cards to insurance. However, the 12.5 favorable cards amount to an average of 25, as the insurance pays 2 to 1. The balance is 30 – 25 = +5 unfavorable situations to the player but favorable to the casino. In this case, there is a house advantage of 5/41 = 12%. Unfortunately, even if we consider averages, taking insurance is a repelling bet for the player.

A formula? It would look complicated symbolically, but it is very easy to follow.

HA = {(R – T) – T*2} / R

where —

• HA = house advantage
• R = cards remaining in the deck
• T = Tens remaining in the deck.

• Axiomatic one, buying (taking) insurance can be a favorable bet for all blackjack players, indeed. Of course, under special circumstances — if you see certain amounts of ten-valued cards on the table. The favorable situations are calculated by the formula above.
But, then again, a dealer natural 21 occurs about 5%- of the time — the insurance alone won't turn the blackjack game entirely in your favor.

3. Calculate Blackjack Double-Down Hands

Up until yours truly wrote such software, total elements in blackjack (i.e. hands) were obtained via simulation. Problem with simulation is incomplete generation. According to by-now famed Ion Saliu's Probability Paradox, only some 63% of possible elements are generated in a simulation of N random cases.

I tested my software a variable number of card decks and various deck compositions. I noticed that decks with lower proportions of ten-valued cards provided higher percentages of potential double-down hands. It is natural, of course, as Tens are the only cards that cannot contribute to a hand to possibly double down. However, the double-down hands provide the most advantageous situations for blackjack player. Indeed, it sounds like "heresy" to all followers of the cult or voodoo ritual of card counting!

I rolled up my sleeves and performed comprehensive calculations of blackjack double-downs (2-card hands). The single deck is mostly covered, but the calculations can be extended to any number of decks.

At the beginning of the deck (shoe): Total combinations of 52 cards taken 2 at a time is C(52, 2) = 1326 hands. Possible 2-card combinations that can be double-down hands:

• 9-value cards AND 2-value cards: 4 9s * 4 2s = 16 two-card possibilities
• 8-value cards AND 2-value cards: 4 8s * 4 2s = 16 two-card configurations
• 8-value cards AND 3-value cards: 4 8s * 4 3s = 16 two-card possibilities
• 7-value cards AND 2-value cards: 4 7s * 4 2s = 16 two-card configurations
• 7-value cards AND 3-value cards: 4 7s * 4 3s = 16 two-card possibilities
• 7-value cards AND 4-value cards: 4 7s * 4 4s = 16 two-card configurations

• 6-value cards AND 3-value cards: 4 6s * 4 3s = 16 two-card configurations
• 6-value cards AND 4-value cards: 4 6s * 4 4s = 16 two-card combinations
• 6-value cards AND 5-value cards: 4 6s * 4 5s = 16 two-card possibilities
• 5-value cards AND 4-value cards: 4 5s * 4 4s = 16 two-card combinations
• 5-value cards AND 5-value cards: C(4, 2) = 6 two-card hands (5 + 5 can appear 6 ways).

• Ace AND 2-value cards: 4 As * 4 2s = 16 two-card combinations
• Ace AND 3-value cards: 4 As * 4 3s = 16 two-card possibilities
• Ace AND 4-value cards: 4 As * 4 4s = 16 two-card hands
• Ace AND 5-value cards: 4 As * 4 5s = 16 two-card possibilities
• Ace AND 6-value cards: 4 As * 4 6s = 16 two-card hands
• Ace AND 7-value cards: 4 As * 4 7s = 16 two-card combinations.

• Total possible 2-card hands in doubling down configuration: 262. Not every configuration can be doubled down (e.g. 4+5 against Dealer's 9 or A+2 against 7).
• We look at a double down blackjack basic strategy chart. Some 42% of the hands ought to be doubled-down (strongly recommended): 262 * 0.42 = 110. That figure represents 8% of total possible 2-hand combinations (1362), or a chance equal to once in 12 hands.

• The chance for double-down situations increases with an increase in tens out over the one third cutoff count. The probability for a natural blackjack decreases also — one reason the traditional plus-count systems anathema the negative counts. But what's lost in naturals is gained in double downs — and then some.
• A sui generis blackjack card-counting strategy was devised by yours truly and it beats the traditionalist plus count systems hands down, as it were.
• Be mindful, however, that nothing beats the The Best Casino Gambling Systems: Blackjack, Roulette, Limited Martingale Betting, Progressions. That's the only way to go, the tao of gambling.

4. Calculate Blackjack Pairs: Strict or Mixed Ten-Cards

Mixed Pairs: All 10-Valued Cards Taken 2 at a Time

 Total STAND Hands (S):           462 = 34.84%
 Total STIFF Hands:               864 = 65.16%

 Total PAIRS (e.g. 2+2 or J+K):   174 = 13.12%
 
 DOUBLE DOWNS (Sums 9, 10, 11):   102 = 7.69%
 BJ Natural (10+A):               64 = 4.83%

 TOTAL 2-CARD BJ HANDS:           1326 

Evidently, there are 13 ranks. Nine ranks (2 to 9 and Ace) consist of 4 cards each (in a single deck). Four ranks (the Tenners) consist of 16 cards. Total of mixed pairs is calculated by the combination formula for every rank. C(4, 2) = 6; 6 * 9 = 54 (for the non-10 cards). The Ten-ranks contribute: C(16, 2) = 120. Total mixed pairs: 54 + 120 = 174. Probability to get a mixed pair: 174 / 1326 = 13%.

Strict Pairs: Only 10+10, J+J, Q+Q, K+K

There are 13 ranks of 4 cards each. Each rank contributes C(4, 2) = 6 pairs. Total strict pairs: 13 * 6 = 78. Probability to get a mixed pair: 78 / 1326 = 5.9%. Total strict pairs = 78 2-card hands (5.9%, but...).

5. Free Blackjack Resources, Basic Strategy, Casino Gambling Systems

• Blackjack Mathematics Probability Odds Basic Strategy Tables Charts.
• The Best Blackjack Basic Strategy: Free Cards, Charts.
  ~ All playing decisions on one page — absolutely the best method of learning Blackjack Basic Strategy (BBS) quickly (guaranteed and also free!)

• Blackjack Gambling System Based on Mathematics of Streaks.
• Deck, Shoe Start, Reversed Card Counting: Blackjack Systems
• Blackjack Card Counting Cult, Deception in Gambling Systems.
• The Best Blackjack Strategy, System Tested with the Best Blackjack Software.
• Reality Blackjack: Real, Fake Odds, House Advantage, Edge.

Back to Forums Index        Socrates Home  Search