markhowy
Blackjack percentage calculations, who knows the real figures?
I was once in Vegas and asked one of the senior management what is the edge to the bank in Blackjack. He said “No one can figure that out, it’s even too complicated for the best mathematicians in the country”.
I laughed, knowingly just taken several thousand dollars with my roulette computer; the same remark had been made to me several years ago when I had suggested that I could beat roulette with a computer!
I received a recent email from someone who has posted in the Blackjack section on here. He has sent me a couple of links to investigate, both of them contradictory. This is not my area, but decided to do a few calculations while listening to music!
I am not the expert in this field, but after 2 hours, came up with the following, correct me if there is any mistake in this ;-)
It is complicated, the main thing that gives the bank its edge is the following>
If you and the dealer have the same value, there is a stand off, no one wins or loses, this is not correct, it is in fact wrong. This is true if 21 or less, but if you are higher, the bank calls bust and takes your money, the bank does not wait to see if he ties or busts higher than this, so the player loses!
Individual players can stand at 14, 15, 16 etc, so the game changes with different strategies or rules for the player. The dealer on the other hand, must follow strict rules, so we can calculate his percentage, and therefore the banks.
The dealer has no choice as to whether he sticks or draws. The dealer’s advantage is that the player is dealt first, if the dealer and player bust over 21, the player loses his bet, not the bank. If the player was not allowed to split pairs or double down and the above was not a governing rule, that is, the busting over 21, then the game would be all fair.
The banks percentage changes with each card, it can go up or down with each card. I will use only a 52 deck of cards, but the process can be adapted to more than one pack, showing the same bankers edge, I assume.
We now need to know what is the combination of all hands the dealer can produce, every permutation and combination.
We produce the following graph.
1326 two card counts in blackjack
Count Number of possibilities
21 - 64
20 - 136
19 - 80
18 - 86
17 - 96
16 - 86
15 - 96
14 - 102
13 - 112
12 - 118
11 - 64
10 - 54
9 - 48
8 - 38
7 - 32
6 - 38
5 - 32
4 - 22
3 - 16
2 - 6
Number of combinations 1326
This includes the player counting the Ace as 1 or 11!
If we calculate the number of fractions that the dealer could bust, we would complicate thinks more, so I have picked the number 169 as a common number to work with pertaining to this game.
Now multiply 1326* 169=224,094
Having 17 or more will show up 462 times out of 1326. So 462/1326*224094=78,078 times.
The dealer must stick on all these hands, so he cannot bust!
Two card counts of 12, 13,14,15,16 will be held by the dealer 514 times.
So 514/1326*224094=86,899.
Since the dealer must draw on 16 or less, he will bust 47,456 times and reach 17-21 in 39410 out of the initial 86866 hands.
Two card counts of 2, 3, 4,5,6,7,8,9,10,11 will appear 350 times, or 350/1326*224094=59150 times.
When the dealer gets any of these counts, he must draw other card/cards, he will therefore bust 17018 times and reach 17-21 in 42132 times out of the 59150 hands.
Now add all the busts together we have 64474
All the non bust hands add up too 159620
Divide the 64474 into the total number of hands 224094 and we find the dealer will bust every 3.47 hands, or approximately 28 hands out of a hundred!
Now, in this calculation, I have assumed a Player following similar rules to the bank, he sticks on 17 or more, but draws if on 16 or less. He at present has not split pairs.
Knowing this we can calculate the exact percentage the player has and the bank has.
If the player is playing the same rules as the dealer, we can simply multiply the fractions together as follows>
64474/224094*64474/224094=4156896676/50218120836
Simply divide the top figure by the bottom one.
He bank has a favourable bust edge of 8.27%
Now we have not yet finished, the bank is collecting this 8.27%, but it has also to pay 3-2 to players holding 21 on their first two cards. On average this occurs 1 in 21 times.
The bank is therefore paying an additional bonus of 2.37%. (Note the stand off, where Dealer and Player have 21 happens every 441 deals, so this percentage we will ignore)
Now deduct this 2.37% from 8.27 bank advantage.
We are left with the answer of 5.90%
This is the banks edge!!!!!!
I wonder if any casino staffs know this, more importantly the fundamentals behind how this figure is produced.
Hope this helps you Folks; this is the last bit of Blackjack mathematics that I touch, so no more questions on the subject, please!
I hope this is a little more friendly than the mathematics at BJ21.com, I have tried to keep it simple.
Speak to you soon. .
Mark Anthony Howe
I was once in Vegas and asked one of the senior management what is the edge to the bank in Blackjack. He said “No one can figure that out, it’s even too complicated for the best mathematicians in the country”.
I laughed, knowingly just taken several thousand dollars with my roulette computer; the same remark had been made to me several years ago when I had suggested that I could beat roulette with a computer!
I received a recent email from someone who has posted in the Blackjack section on here. He has sent me a couple of links to investigate, both of them contradictory. This is not my area, but decided to do a few calculations while listening to music!
I am not the expert in this field, but after 2 hours, came up with the following, correct me if there is any mistake in this ;-)
It is complicated, the main thing that gives the bank its edge is the following>
If you and the dealer have the same value, there is a stand off, no one wins or loses, this is not correct, it is in fact wrong. This is true if 21 or less, but if you are higher, the bank calls bust and takes your money, the bank does not wait to see if he ties or busts higher than this, so the player loses!
Individual players can stand at 14, 15, 16 etc, so the game changes with different strategies or rules for the player. The dealer on the other hand, must follow strict rules, so we can calculate his percentage, and therefore the banks.
The dealer has no choice as to whether he sticks or draws. The dealer’s advantage is that the player is dealt first, if the dealer and player bust over 21, the player loses his bet, not the bank. If the player was not allowed to split pairs or double down and the above was not a governing rule, that is, the busting over 21, then the game would be all fair.
The banks percentage changes with each card, it can go up or down with each card. I will use only a 52 deck of cards, but the process can be adapted to more than one pack, showing the same bankers edge, I assume.
We now need to know what is the combination of all hands the dealer can produce, every permutation and combination.
We produce the following graph.
1326 two card counts in blackjack
Count Number of possibilities
21 - 64
20 - 136
19 - 80
18 - 86
17 - 96
16 - 86
15 - 96
14 - 102
13 - 112
12 - 118
11 - 64
10 - 54
9 - 48
8 - 38
7 - 32
6 - 38
5 - 32
4 - 22
3 - 16
2 - 6
Number of combinations 1326
This includes the player counting the Ace as 1 or 11!
If we calculate the number of fractions that the dealer could bust, we would complicate thinks more, so I have picked the number 169 as a common number to work with pertaining to this game.
Now multiply 1326* 169=224,094
Having 17 or more will show up 462 times out of 1326. So 462/1326*224094=78,078 times.
The dealer must stick on all these hands, so he cannot bust!
Two card counts of 12, 13,14,15,16 will be held by the dealer 514 times.
So 514/1326*224094=86,899.
Since the dealer must draw on 16 or less, he will bust 47,456 times and reach 17-21 in 39410 out of the initial 86866 hands.
Two card counts of 2, 3, 4,5,6,7,8,9,10,11 will appear 350 times, or 350/1326*224094=59150 times.
When the dealer gets any of these counts, he must draw other card/cards, he will therefore bust 17018 times and reach 17-21 in 42132 times out of the 59150 hands.
Now add all the busts together we have 64474
All the non bust hands add up too 159620
Divide the 64474 into the total number of hands 224094 and we find the dealer will bust every 3.47 hands, or approximately 28 hands out of a hundred!
Now, in this calculation, I have assumed a Player following similar rules to the bank, he sticks on 17 or more, but draws if on 16 or less. He at present has not split pairs.
Knowing this we can calculate the exact percentage the player has and the bank has.
If the player is playing the same rules as the dealer, we can simply multiply the fractions together as follows>
64474/224094*64474/224094=4156896676/50218120836
Simply divide the top figure by the bottom one.
He bank has a favourable bust edge of 8.27%
Now we have not yet finished, the bank is collecting this 8.27%, but it has also to pay 3-2 to players holding 21 on their first two cards. On average this occurs 1 in 21 times.
The bank is therefore paying an additional bonus of 2.37%. (Note the stand off, where Dealer and Player have 21 happens every 441 deals, so this percentage we will ignore)
Now deduct this 2.37% from 8.27 bank advantage.
We are left with the answer of 5.90%
This is the banks edge!!!!!!
I wonder if any casino staffs know this, more importantly the fundamentals behind how this figure is produced.
Hope this helps you Folks; this is the last bit of Blackjack mathematics that I touch, so no more questions on the subject, please!
I hope this is a little more friendly than the mathematics at BJ21.com, I have tried to keep it simple.
Speak to you soon. .
Mark Anthony Howe