Blackjack Dollar Sevens

The house edge of Dollar Sevens is calculated on the premise that there are 8 decks in the shoe ie. there are a total of 52 x 8 = 416 cards of which 8 x 4 = 32 are sevens.

First we need to calculate the expected return of the base game which consists of the following 4 outcomes

2 sevens not suited
2 sevens of the same suit
3 sevens not suited
3 sevens of the same suit

and consider the probability of each of these events occurring.

2 Sevens not suited

The probability of the first card being a seven is 32/416 because there are 32 ways of choosing a seven out of 416 cards. After the first card has been dealt there are 415 different ways of choosing the second card. As there are 8 sevens of each suit, after the first seven has been dealt, there are 24 = 32 – 8 different ways of choosing the second seven which is of a different suit than the first. Therefore the probability of the second seven being dealt of a different suit than the first is 24/415. There are now a total of 414 cards left to choose from and 384 = 416 – 32 of these are non-seven cards so that the probability of the third card not being a seven is 384/414.

2 Sevens suited

The probability of the first card being a seven is exactly the same as with the previous result of non-suited sevens ie. 32/416. Once the first seven has been dealt, there are 7 sevens remaining in the deck of the same suit and a total of 415 cards to choose from so that the probability of being dealt another seven of the same suit as the first is 7/415. There are 384 different ways that the third card dealt can be a non-seven out of a possible 414 card left in the deck so that the probability of this occurring is 384/414.

3 Sevens not suited

Consider each of the three different ways that one can achieve a hand which contains three sevens which are not suited:-

– The first two sevens are the same suit and the third is of a different suit.

The probability of the first card being a seven is again 32/416. There are 7 ways to choose the second 7 card so that it is of the same suit as the first, and a total of 415 cards remaining in the deck to choose from so that the probability of being dealt another seven of the same suit as the first is 7/415. In order to satisfy the criteria of the 3 sevens not being of the same suit the third card drawn must be seven of a different suit than the first two. There are 24 ways of choosing a seven which is of a different suit than the first two cards dealt and there are 414 cards remaining in the deck so that the probability of the third card being a seven but not of the same suit as the first two chosen is 24/414.

– The first and third seven are of the same suit but the second seven is of a different suit 

The probability of the first card being any seven is 32/416. After the first seven has been dealt, there are 24 = 32 – 8 different ways of choosing the second seven which is of a different suit than the first. Therefore the probability of the second seven being dealt of a different suit than the first is 24/415. There are 7 ways to choose the third 7 card so that it is of the same suit as the first, and a total of 414 cards remaining in the deck to choose from so that the probability of being dealt a third seven of the same suit as the first is 7/414.

– The suit of the first seven is different from the second and third (although the second and third seven may be of the same suit)

The probability of the first card being any seven is 32/416. After the first seven has been dealt, there are 24 = 32 – 8 different ways of choosing the second seven which is of a different suit than the first. Therefore the probability of the second seven being dealt of a different suit than the first is 24/415. The suit of the third seven may be the same or different from the suit of the second card ensuring however that it is not the same as the suit of the first. Therefore there are 23 = 24 – 1(card 2) ways of choosing the third seven which is not of the same suit as the first seven, out of a possible deck of 414 cards. The probability of this occurring is 23/414.

3 Sevens suited

The probability of the first card being any seven is 32/416. Once the first seven has been dealt, there are 7 sevens remaining in the deck of the same suit and a total of 415 cards to choose from so that the probability of being dealt another seven of the same suit as the first is 7/415. If two sevens of the same suit have been dealt then there are 6 sevens remaining in the deck of the same suit as the first two out of a possible 414 cards, so that the probability of this occurring is 6/414.

The overall probability of achieving each of the above hands is the product of the probabilities of each of three cards dealt. For example, the probability of achieving 3 suited sevens equals 32/416 x 7/415 x 6/414 = 0.00188%. Note that the probability of achieving three sevens that are not suited is the sum of the individual products associated with each of the three different options ie.

prob. of 3 sevens not suited = 32/416 x 7/415 x 24/414
+ 32/416 x 24/415 x 7/414
+ 32/416 x 24/415 x 23/414 = 0.040%.

The expected return to the player E[X] for each of these results is arrived at by multiplying the probability of the result occurring by the payout entitlement for each of these results. For example the expected return to the player for a hand containing 2 suited sevens is (32/416 x 7/415 x 384/414) x 150 = 18.05%. This basically means that in the long term, of every dollar wagered by a player on Dollar Sevens he/she will expect to recoup approximately 18 cents on all hands with 2 suited sevens. These results are summarised in the top part of the table below where P(1), P(2), and P(3) indicate the probabilities of drawing the first, second and third card respectively. The overall probability of achieving the hand is represented by P(W) which is the product of P(1), P(2), and P(3).

In addition to the ordinary payout schedule, every player who achieves three suited sevens receives an automatic entry form to a Dollar Sevens Tournament. A Tournament will be held when we have identified 49 participants and based on our current volume of Blackjack play it is expected that this will occur every three months or so. The Tournament entrants will then have the opportunity to win a $250,000 first prize.

As the Tournament prizes are guaranteed, the cost of these must be incorporated into the calculation of the overall game house edge.

Once we have filled a Tournament with the 49 players who achieved three suited sevens each entrant will receive $1,000 in cash chips at the commencement of the Tournament heats. The cost of these chips are incorporated into the calculation of player return and hence house edge. The probability of a Dollar Seven player gaining entry to a Tournament and hence qualifying for the $1,000 in chips is effectively the same as the probability of achieving 3 suited sevens as this is the only way that Tournament entry can be achieved. Therefore the probability of receiving $1,000 in chips is 32/416 x 7/415 x 6/414 = 0.00188% and therefore the expected return to player E[X] for this component of the Tournament is the probability of this occurring multiplied by the payout ie. 0.00188% x $1,000 = 1.88%.

The rest of the Tournament prize table is also incorporated into the calculation of house edge. Consider firstly the $250,000 first prize and the impact that this prize has on the expected return to the player. Once a Dollar Sevens player has gained entry to a Tournament he/she has a 1 in 49 chance of winning the $250,000 first prize as there are 49 entrants. Therefore the probability of winning $250,000 is equal to the probability of gaining entry to the Tournament multiplied by the probability of winning the Tournament ie, 0.00188% x 1/49 = 0.0004%.

The expected return to player for this component is 0.0004% x $250,000 = 9.59%

Similar calculations can be performed for each of the other Tournament prizes. There is a 1 in 49 chance of winning the second prize in the Tournament so that once again the probability of winning $50,000 is equal to the probability of gaining entry to the Tournament multiplied by the probability of winning second place in the Tournament, As there is only one second place the overall probability of winning $50,000 is 0.00188% x 1/49 = 0.0004%.

The expected return to player for this component is 0.0004% x $50,000 = 1.92%

The remainder of the Tournament prizes are treated in the exact same manner and are summarised in the above table. The overall contribution to the return to player for the Tournament is 16.08% = 1.88% + 9.59% +1.92% + 0.96% + 0.58% + 1.15%.

The probability of the player not achieving any winning result is equal to 1 – probability of being dealt a winning hand ie the probability of being dealt a losing hand is equal to 1 – (0.413% + 0.12% +0.04% +0.00188%) = 99.425%. The expected return to the player for a losing hand is equal to the probability of this occurring multiplied by the amount wagered ie the expected return = 99.425% x (-1) = -99.425%.

To determine the overall return to player of all the different components of the game one needs to sum over the individual returns for all the possible outcomes including the Tournament prizes. The sum of the return to player is 89.83% as indicated in the right hand column of the above table. If 89.83% of every dollar wagered is returned to the player then the house keeps 1 – 89.83% = 10.17% of the total turnover meaning that the house advantage on Dollar Sevens is 10.17%.