GAM 470
Probabilities in Blackjack Side Bets Pt 2
21+3
This is a side bet based on the player’s first two cards and the dealer’s up card. I’ve seen 2 and 6 decks used in the casinos but let’s assume six decks here. Following are the possible wins and the number of combinations.
Straight Flush
There are 4 possible suits. With three cards there are 12 possible spans for a straight flush (including the "mini royal") A-3 to Q-A. Once the span and the suit are chosen there are six cards in the shoe for each card in the straight flush. So the total combinations are 4*12*6^{3} = 10,368.
Three of a Kind
There are 13 possible ranks for the three of a kind. One the rank is chosen there are 4*6=24 in the shoe from. There are combin(24,3)=2024 ways to choose 3 cards out of 24. So the total combinations are 13*2024 = 26,312.
Straight
There are 12 possible spans for the straight. For each card in the span there are 24 cards to choose from. So there are 12*24^{3} = 165,888 ways to get three cards in a row. However 10,368 will result in a straight flush. So the total number of ways to get a straight are 165,888-10,368 = 155,520.
Flush
There are 4 possible suits for the flush. Once a suit is chosen there are 13*6 = 78 cards of that suit in the shoe. There are combin(78,3) = 76,076 ways to draw 3 cards out of 78. So there are 4*76,076 = 304,304 ways to draw three cards of the same suit. However some of those ways will be drawing three cards of the same rank, resulting in a hand which is both a flush and a three of a kind. This hand would be scored as a three of a kind, so must be subtracted from this group. There are 52 cards in the deck. Once a card is chosen there are combin(6,3)=20 ways to pick 3 cards out of 6. So the number of suited three of a kinds is 52*20=1040. 10,368 suited combinations are straight flushes, which must also be subtracted. So the number of flushes is 304,304 – 1040 – 10,368 = 292,896.
Pair
There are 13 possible ranks for the pair. Then there are 12 ranks left for the singleton. Once the rank is chosen there are combin(24,2) = 276 ways to choose 2 cards out of the 24 of that rank in the shoe. There are 24 ways to choose the singleton, once the rank is determined. So the total number of ways to get two of the same rank is 13*12*276*24 = 1,033,344.
However it is possible the player gets a pair, which is also a higher paying flush. These hands must be subtracted out. There are still 13 ways to choose the rank of the pair and 12 ways for the rank of the singleton. Then there are 4 ways to choose the suit. Then there are combin(6,2) = 15 ways to choose the two cards of the pair and 6 ways for the singleton. So the number of ways to get a hand which is both a pair and a flush is 13*12*4*15*6 = 56,160. Thus the number of ways to get a pair, which is not also a flush, is 1,033,344 – 56,160 = 977,184.
Nothing
For the player to get nothing he must have three different ranks. There are combin(13,3) = 286 ways to choose 3 ranks out of 13. However 12 of those will result in 3 consecutive cards, giving us a straight. So the number of ways to pick 3 ranks out of 13, that are not consecutive, is 286-12 = 274. Once three ranks are chosen there are 24 possible cards in the shoe for each one. Thus there are 24^{3} = 13,824 ways to pick the specific card once the ranks have been chosen. So the total number of ways to pick 3 different ranks is 274*13,824 = 3,787,776.
However some of these combinations will result in a flush. There are still 274 ways to pick the ranks. There are 4 possible suits for a flush. With three different ranks there are 6 cards to choose from for each card. So there are 274*4*6^{3} = 236,736 ways to make a flush with 3 different ranks.
Thus the total number of ways to get nothing is 3,787,776 – 236,736 = 3,551,040.
All hands of a flush or higher pay 9 to 1. A pair or nothing lose. Following is the return table.
Hand |
Combinations |
Pays |
Return |
Straight flush |
10368 |
9 |
93312 |
Three of a kind |
26312 |
9 |
236808 |
Straight |
155520 |
9 |
1399680 |
Flush |
292896 |
9 |
2636064 |
Pair |
977184 |
-1 |
-977184 |
Nothing |
3551040 |
-1 |
-3551040 |
Total |
5013320 |
-162360 |
So the expected value is –163360/5013320 = -3.239%.
Lucky Ladies
Lucky Ladies is a side bet based mostly on the player’s first two cards. However the highest hand also requires the dealer to have a blackjack. Following are how to calculate the number of combinations in a 6-deck game.
Player queen of hearts pair and dealer has a blackjack
There are six queen of hearts in the shoe. There are combin(6,2)=15 ways to pick 2 out of the six. So there are 15 ways the player can achieve his half of this hand.
There are 4*6=24 aces in the deck for the dealer’s blackjack. With 2 queens removed from the deck there are 16*6-2 = 94 ten points cards left. So the number of ways the dealer can get a blackjack is 24*94 = 2256.
So the total ways the player can get a queen of hearts pair and the dealer a blackjack is 15*2256 = 33,840.
Player queen of hearts pair and dealer doesn’t have a blackjack
There are still 15 ways the dealer can get a queen of hearts pair.
There are a total of 52*6-2 = 310 cards left. There are combin(310,2) = 47,895 ways the 2 cards out of 310 can be chosen for the dealer. However 2256 ways will result in a blackjack. So 47,895-2256 = 45,639 will result in no dealer blackjack.
So the total number of ways the player can get a queen of hearts pair and the dealer does not have a blackjack is 15*45,639 = 684,585.
Matched 20
A matched 20 is a player hand that totals 20, in which the two cards are the same in rank and suit. This rules out A/9 twenties because the ranks are different. The dealer’s hand can be anything.
There are 4 possible ranks and 4 possible suits for the match 20, or 16 cards per deck. However one of these cards is the queen of hearts, which would result in a higher paying hand. So there are 15 cards left in a deck which can be used for the matched 20. Once a card and suit have been chosen there are combin(6,2)=15 ways to choose 2 of them from the 6 in the shoe. So the total number of ways the player can get a matched 20 is 15*15 = 225.
The number of possible dealer combinations is combin(52*6-2,2) = combin(310,2) = 47,895. So the total number of combinations is 225*47895 = 10,776,375.
Suited 20
There are two types of suited 20’s: (1) two 10-point cards of different ranks and (2) an ace and 9.
Let’s evaluate the 10/10 twenty first. There are 4 ranks worth 10 points in the shoe and we much choose 2 of them. There are combin(4,2)=6 ways to choose 2 ranks out of 4. Then there are 4 suits to choose from. Once two specific cards have been chosen there are six of each in the shoe to choose from. So there are 6*4*6^{2} = 864 ways the player can get a type (1) suited 20.
For the A/9 twenty there are also 4 suits to choose from. Then there are six of each card in the shoe. So there are 4*6^{2} = 144 ways the player can get a suited A/9.
So the total number of ways the player can get a suited (but not ranked) twenty is 864+144 = 1008.
There are still combin(310,2) = 47,895 dealer combinations.
So the total player & dealer combinations for a suited 20 is 1008*47,895 = 48,278,160.
Unsuited 20
Again, there are two types of twenties, the 10/10 and A/9. Either way there are combin(4,2)=6 ways to pick 2 suits out of 4. Once two suits have been chosen there are 4 tens in each deck of that suit and six decks total to choose from. In other words there are 24 10-point cards in the shoe of any given suit. So there are 6*24^{2} = 3456 ways the player can get an unsuited 10/10 twenty.
For the A/9 twenty let’s arbitrarily say the two suits are spades and clubs. Let’s also arbitrary decide to assign the spade to the ace and the club to the 9. Once the ranks and suits have been chosen, and the ranks assigned to the suits, there are 6 of each card to choose from in the shoe. However in the example the spade could have just as easily been assigned to the 9 and the club to the ace. So we must multiply by 2 because there are two ways to arrange the suits to the ranks. Overall there are 6*2*6^{2 = }432 ways the player can get an unsuited A/9 twenty.
There are 3456+432 = 3888 total ways the player can get an unsuited 20. Again there are combin(310,2)=47,895 ways the dealer can choose 2 cards out of 310. So the total player & dealer combinations are 3888*47895 = 165,525,120.
Non-20
There are various ways to get a total that does not equal as follows:
Both ranks in 2 to 8: There are 7 ranks in the span from 2 to 8, so 7*4=28 per deck, or 28*6=168 in the whole shoe. There are combin(168,2)=14,028 ways to pick any 2 of them.
One rank 2 to 8, the other 9-A: There are 6 ranks 9 to ace, 6*4=24 per deck, and 24*6=144 in the whole shoe. So there are 168 ways to pick the 2-8 card and 144 ways to pick the 9-A card, for a total of 168*144 = 24,192 ways to pick a 2-8,9-A hand.
Two aces: There are 24 aces in the whole shoe so combin(24,2)=276 ways to pick two of them.
Two nines: There are 24 nines in the whole shoe so combin(24,2)=276 ways to pick two of them.
10-pt/A or 10-pt/9: There are 16*6=96 10-point cards in the whole shoe. There are 8 nines or aces per deck so 8*6=48 in the whole shoe. So there are 96*48=4608 ways to pick one 10-point card and one ace or 9.
So the total number of ways the player can get anything except a 20 is 14,028 + 24,192 + 2*276 + 4608 = 43,380.
There are still combin(310,2) = 47,895 dealer combinations.
So the total player & dealer combinations for a non-20 is 43,380*47,895 = 2,077,685,100.
The following return table shows the total return of this game over all combinations.
Player has |
Dealer has |
Player combin. |
Dealer combin. |
Total combinations |
Pays |
Return |
Queen of hearts pair |
Blackjack |
15 |
2,256 |
33,840 |
1,000 |
33,840,000 |
Queen of hearts pair |
Not blackjack |
15 |
45,639 |
684,585 |
125 |
85,573,125 |
Matched 20 |
Anything |
225 |
47,895 |
10,776,375 |
19 |
204,751,125 |
Suited 20 |
Anything |
1,008 |
47,895 |
48,278,160 |
9 |
434,503,440 |
Unsuited 20 |
Anything |
3888 |
47,895 |
186,215,760 |
4 |
744,863,040 |
Non-20 |
Anything |
43,380 |
47,895 |
2,077,685,100 |
-1 |
-2,077,685,100 |
Total |
2,323,673,820 |
-574,154,370 |
Expected value = -574,154,370 / 2,323,673,820= -0.247089056 = -24.71%.