Category Archives: Concepts

Concepts 16: Baccarat – Effects of Card Removal

Hi friends,

This topic follows the study on the most beneficial cards to the Player, Banker and Tie wagers in Baccarat.

The 3 tables below show the effect on the probabilities of the Player, Banker and Tie as 1, 2 and 3 cards are removed in isolation from each card value. Do take note of the changes in probability as each card is removed and their relationship to the respective wager.

Player

Card Value: Permutations Player Win % Compared to Other Wagers Player Win % by Card Value Effect of 1 Card Removed Effect of 2 Cards Removed Effect of 3 Cards Removed
4    1,036,330,148,130,820 44.9220% 7.7436% –                   0.0000435175 –                     0.0000870460 –                     0.0001305855
0    1,035,362,791,243,780 44.8801% 7.7363% –                   0.0000373810 –                     0.0000749287 –                     0.0001126377
J    1,035,362,791,243,780 44.8801% 7.7363% –                   0.0000373810 –                     0.0000749287 –                     0.0001126377
Q    1,035,362,791,243,780 44.8801% 7.7363% –                   0.0000373810 –                     0.0000749287 –                     0.0001126377
K    1,035,362,791,243,780 44.8801% 7.7363% –                   0.0000373810 –                     0.0000749287 –                     0.0001126377
1    1,034,132,990,550,020 44.8268% 7.7272% –                   0.0000295798 –                     0.0000594307 –                     0.0000895596
3    1,032,891,882,528,770 44.7730% 7.7179% –                   0.0000217068 –                     0.0000440304 –                     0.0000669774
2    1,031,651,228,684,290 44.7192% 7.7086% –                   0.0000138368 –                     0.0000279871 –                     0.0000424652
9    1,031,234,689,822,720 44.7012% 7.7055% –                   0.0000111945 –                     0.0000236585 –                     0.0000374332
8    1,030,998,408,818,690 44.6909% 7.7037% –                   0.0000096956 –                     0.0000214609 –                     0.0000353579
5    1,020,528,719,052,800 44.2371% 7.6255%                      0.0000567188                        0.0001128483                        0.0001683503
7    1,013,469,785,956,350 43.9311% 7.5728%                      0.0001014971                        0.0002024677                        0.0003028843
6    1,010,420,677,033,980 43.7989% 7.5500%                      0.0001208391                        0.0002418228                        0.0003629233

Banker

Card Value: Permutations Banker Win % Compared to Other Wagers Banker Win % by Card Value Effect of 1 Card Removed Effect of 2 Cards Removed Effect of 3 Cards Removed
8    1,067,894,055,137,280 46.2902% 7.7645% –                   0.0000629994 –                     0.0001283117 –                     0.0001959930
9    1,063,659,370,418,180 46.1067% 7.7337% –                   0.0000361367 –                     0.0000740128 –                     0.0001136737
5    1,062,272,823,181,310 46.0466% 7.7236% –                   0.0000273411 –                     0.0000544194 –                     0.0000812294
0    1,061,053,355,229,180 45.9937% 7.7148% –                   0.0000196054 –                     0.0000395286 –                     0.0000597675
J    1,061,053,355,229,180 45.9937% 7.7148% –                   0.0000196054 –                     0.0000395286 –                     0.0000597675
Q    1,061,053,355,229,180 45.9937% 7.7148% –                   0.0000196054 –                     0.0000395286 –                     0.0000597675
K    1,061,053,355,229,180 45.9937% 7.7148% –                   0.0000196054 –                     0.0000395286 –                     0.0000597675
6    1,056,701,970,653,180 45.8051% 7.6831%                      0.0000079976                        0.0000143452                        0.0000189833
1    1,055,563,635,941,380 45.7557% 7.6749%                      0.0000152186                        0.0000304640                        0.0000457306
7    1,054,846,110,916,610 45.7246% 7.6696%                      0.0000197702                        0.0000379094                        0.0000543471
2    1,051,591,007,948,800 45.5835% 7.6460%                      0.0000404189                        0.0000808770                        0.0001213820
3    1,050,791,343,448,060 45.5489% 7.6402%                      0.0000454916                        0.0000893156                        0.0001314284
4    1,045,981,660,065,790 45.3404% 7.6052%                      0.0000760018                        0.0001513286                        0.0002259417

Tie

Card Value: Permutations Tie % Compared to Other Wagers Tie % by Card Value Effect of 1 Card Removed Effect of 2 Cards Removed Effect of 3 Cards Removed
6       239,830,402,545,152 10.3960% 8.4040% –                   0.0001288367 –                     0.0002561680 –                     0.0003819067
7       238,637,153,359,360 10.3443% 8.3622% –                   0.0001212674 –                     0.0002403772 –                     0.0003572314
4       224,641,242,035,712 9.7376% 7.8718% –                   0.0000324844 –                     0.0000642826 –                     0.0000953562
5       224,151,507,998,208 9.7163% 7.8546% –                   0.0000293777 –                     0.0000584289 –                     0.0000871209
2       223,710,813,599,232 9.6972% 7.8391% –                   0.0000265822 –                     0.0000528899 –                     0.0000789169
3       223,269,824,255,488 9.6781% 7.8237% –                   0.0000237848 –                     0.0000452853 –                     0.0000644510
1       217,256,423,740,928 9.4175% 7.6130%                      0.0000143612                        0.0000289667                        0.0000438290
9       212,058,989,991,424 9.1922% 7.4309%                      0.0000473311                        0.0000976713                        0.0001511069
0       210,536,903,759,360 9.1262% 7.3775%                      0.0000569865                        0.0001144573                        0.0001724053
J       210,536,903,759,360 9.1262% 7.3775%                      0.0000569865                        0.0001144573                        0.0001724053
Q       210,536,903,759,360 9.1262% 7.3775%                      0.0000569865                        0.0001144573                        0.0001724053
K       210,536,903,759,360 9.1262% 7.3775%                      0.0000569865                        0.0001144573                        0.0001724053
8       208,060,586,276,352 9.0188% 7.2907%                      0.0000726950                        0.0001497726                        0.0002313508

In summary, you will notice that shoes with a higher number of certain card values remaining will lead to a better probability on specific wagers. The following table shows the simplified effect of removal of each card value on the respective probabilities of the Player, Banker and Tie wagers:-​

Effect of Removal on Probability by Card Value
Wager 1 2 3 4 5 6 7 8 9 0
Player  – + + +
Banker + + + + + (- from removal of card 10) + (-from removal of card 20)
Tie +

(+ from Removal of card 20)

+ + +

In other words:

  1. For the Player, it would be advantageous to have as many 5s, 6s and 7s removed from the shoe as possible and to keep the rest in.
  2. For the Banker, removal of the 1s through to 4s and 6s and 7s would be advantageous. Do note the peculiarities at card 10 for 6 and card 20 for 7.
  3. For Tie, removal of the 1s and 8s through to Ks would be advantageous. Do note the peculiarity at card 20 for 3.

The 3 charts below show the effect on probability as 1 to 32 cards of each value are removed.

Player

Concepts 16 Player

Banker

Concepts 16 Banker

Tie

Concepts 16 Tie

Good luck!

Concepts 15: Most Beneficial Cards for Each Wager

Friends,

The following 3 tables show the most beneficial cards for each wager.
This was calculated by observing the number of times the presence of cards of these values led to a Player, Banker or Tie result.
They are sorted in descending order of effect by card value. Some of the results are intuitive, others may surprise.

Column 3: Player Win % Compared to Other Wagers compares the effect the card value has on a wager as compared to the other wagers, in this instance, Banker and Tie.

Column 4: Player Win % by Card Value compares that card value’s effect as compared to all other card values for that particular wager.

Player

Card Value: Permutations Player Win % Compared to Other Wagers Player Win % by Card Value
4    1,036,330,148,130,820 44.9220% 7.7436%
0    1,035,362,791,243,780 44.8801% 7.7363%
J    1,035,362,791,243,780 44.8801% 7.7363%
Q    1,035,362,791,243,780 44.8801% 7.7363%
K    1,035,362,791,243,780 44.8801% 7.7363%
1    1,034,132,990,550,020 44.8268% 7.7272%
3    1,032,891,882,528,770 44.7730% 7.7179%
2    1,031,651,228,684,290 44.7192% 7.7086%
9    1,031,234,689,822,720 44.7012% 7.7055%
8    1,030,998,408,818,690 44.6909% 7.7037%
5    1,020,528,719,052,800 44.2371% 7.6255%
7    1,013,469,785,956,350 43.9311% 7.5728%
6    1,010,420,677,033,980 43.7989% 7.5500%

Banker

Card Value: Permutations Banker Win % Compared to Other Wagers Banker Win % by Card Value
8    1,067,894,055,137,280 46.2902% 7.7645%
9    1,063,659,370,418,180 46.1067% 7.7337%
5    1,062,272,823,181,310 46.0466% 7.7236%
0    1,061,053,355,229,180 45.9937% 7.7148%
J    1,061,053,355,229,180 45.9937% 7.7148%
Q    1,061,053,355,229,180 45.9937% 7.7148%
K    1,061,053,355,229,180 45.9937% 7.7148%
6    1,056,701,970,653,180 45.8051% 7.6831%
1    1,055,563,635,941,380 45.7557% 7.6749%
7    1,054,846,110,916,610 45.7246% 7.6696%
2    1,051,591,007,948,800 45.5835% 7.6460%
3    1,050,791,343,448,060 45.5489% 7.6402%
4    1,045,981,660,065,790 45.3404% 7.6052%

Tie

Card Value: Permutations Tie % Compared to Other Wagers Tie % by Card Value
6       239,830,402,545,152 10.3960% 8.4040%
7       238,637,153,359,360 10.3443% 8.3622%
4       224,641,242,035,712 9.7376% 7.8718%
5       224,151,507,998,208 9.7163% 7.8546%
2       223,710,813,599,232 9.6972% 7.8391%
3       223,269,824,255,488 9.6781% 7.8237%
1       217,256,423,740,928 9.4175% 7.6130%
9       212,058,989,991,424 9.1922% 7.4309%
0       210,536,903,759,360 9.1262% 7.3775%
J       210,536,903,759,360 9.1262% 7.3775%
Q       210,536,903,759,360 9.1262% 7.3775%
K       210,536,903,759,360 9.1262% 7.3775%
8       208,060,586,276,352 9.0188% 7.2907%

 

Concepts 14: Baccarat 1st Card Advantage

All right folks.

First off, please understand that having a 1st-card advantage means that the player has somehow found out what the first card of the as-yet undealed coup is.

This is at worst cheating or at best, if the dealer or procedures are weak, advantage play.

1st-card advantage is essentially a study of all the possible card permutations that start with Ace through to King that result in a Player, Banker or Tie result from the remaining cards in the shoe.

The 2 tables below show that knowing that the 1st card is a 1-5 or 10-K, would give a small advantage on the Banker wager. Knowledge of the 1st card being 6-9 would give an increasing advantage on the Player wager, with knowing a 1st card as being 9 giving a 21% advantage on the Player wager.

Below is a table that shows the probabilities before the start of the shoe:

Card Value: Player Win Banker Win Tie Player Win % Banker Win % Tie % Player Exp Banker Exp Tie Exp
1           159,747,170,699,264           189,857,873,692,672          34,887,130,646,784 41.5476% 49.3789% 9.0736% -0.0783 0.0536 -0.1834
2           160,408,944,947,200           188,725,567,889,408          35,357,662,202,112 41.7197% 49.0844% 9.1959% -0.0736 0.0491 -0.1724
3           161,429,794,906,112           187,253,976,461,312          35,808,403,671,296 41.9852% 48.7016% 9.3132% -0.0672 0.0428 -0.1618
4           162,751,823,355,904           184,376,516,966,400          37,363,834,716,416 42.3290% 47.9533% 9.7177% -0.0562 0.0323 -0.1254
5           166,618,505,969,664           179,969,323,315,200          37,904,345,753,856 43.3347% 46.8070% 9.8583% -0.0347 0.0113 -0.1128
6           173,075,882,563,584           170,168,183,390,208          41,248,109,084,928 45.0141% 44.2579% 10.7279% 0.0076 -0.0297 -0.0345
7           185,770,147,510,272           157,451,915,216,896          41,270,112,311,552 48.3157% 40.9506% 10.7337% 0.0737 -0.0941 -0.0340
8           206,999,120,039,936           140,506,916,139,008          36,986,138,859,776 53.8370% 36.5435% 9.6195% 0.1729 -0.1912 -0.1342
9           215,108,869,548,032           132,334,879,180,800          37,048,426,309,888 55.9462% 34.4181% 9.6357% 0.2153 -0.2325 -0.1328
0           638,608,023,052,288           761,607,414,185,984        137,753,262,916,608 41.5228% 49.5203% 8.9568% -0.0800 0.0552 -0.1939

This table shows the same 16 cards in for each card value, so technically, halfway through the shoe:

Card Value: Player Win Banker Win Tie Player Win % Banker Win % Tie % Player Exp Banker Exp Tie Exp
1                2,404,712,210,944                2,862,066,979,328                525,341,662,848 41.5170% 49.4131% 9.0699% -0.0790 0.0543 -0.1837
2                2,414,405,553,152                2,845,665,957,888                532,049,342,080 41.6843% 49.1299% 9.1857% -0.0745 0.0499 -0.1733
3                2,427,320,125,440                2,826,650,029,056                538,150,698,624 41.9073% 48.8016% 9.2911% -0.0689 0.0445 -0.1638
4                2,441,864,919,040                2,787,715,898,880                562,540,035,200 42.1584% 48.1294% 9.7122% -0.0597 0.0356 -0.1259
5                2,510,828,003,328                2,709,937,753,088                571,355,096,704 43.3490% 46.7866% 9.8644% -0.0344 0.0110 -0.1122
6                2,609,779,693,056                2,562,217,828,352                620,123,331,712 45.0574% 44.2363% 10.7063% 0.0082 -0.0303 -0.0364
7                2,801,721,069,056                2,369,605,165,568                620,794,618,496 48.3712% 40.9108% 10.7179% 0.0746 -0.0951 -0.0354
8                3,123,172,254,720                2,113,348,414,464                555,600,183,936 53.9210% 36.4866% 9.5923% 0.1743 -0.1926 -0.1367
9                3,240,895,371,776                1,993,736,233,472                557,489,247,872 55.9535% 34.4215% 9.6250% 0.2153 -0.2325 -0.1338
0                9,633,645,025,280              11,472,680,607,744            2,062,157,779,456 41.5808% 49.5185% 8.9007% -0.0794 0.0546 -0.1989

Baccarat: Conditions for Positive Expectation on Side Wagers

In simulating different conditions of a Baccarat shoe, the following instances were observed:

  1. Wager refers to the wager type studied – Pairs, Wins on Natural 9s (WN9) and Wins on Natural 8s (WN8).
  2. Win % Probability indicates the percentage probability of that result occurring.
  3. Expectation, based on 11:1 for Pairs and 8:1 for Wins on Natural 8s and 9s, shows the expectation (player’s edge) of the respective wagers with the state of the shoe on the right.
  4. 1 – K indicates how many cards of that value have been dealt.
Wager Win % Probability Expectation 1 2 3 4 5 6 7 8 9 0 J Q K
Pairs 8.66% 3.9% 5 23 23 23 23 23 23 23 23 23 23 23 23
Pairs 25.02% 200.2% 5 5 30 30 30 30 30 30 30 30 30 30 30
WN9 14.27% 28.4% 25 25 25 25 25 25 25 25 5 5 5 5 5
WN8 13.21% 18.9% 25 25 25 25 25 25 25 5 25 5 5 5 5

While sufficient randomization would have occurred with the 26-card cut at the end of the shoe as well as with standard burn-card procedures; this table shows that while only under specific circumstances, it IS possible for the expectation of a side wager to turn against the House and towards the player.

These 4 examples are just some of many possible states.

In the first Pair example, we see a modest 3.9% expectation for the Player, meaning that the Player is expected to win 3.9 cents on the dollar. As a means of comparison, the 2nd Pair example shows an extreme expectation of 2.002 dollars for the player per dollar wagered.

Food for thought and perhaps further study.

Good luck.

Concepts 13: Baccarat Vs. Non-Commission Baccarat – Why does a Banker Win on 6 Pay Half?

Why does a Banker win on 6 pay half?

If you have ever wondered about this, you are not alone.

This table shows the Probability and House Edge for Baccarat (where Banker wins pay 0.95:1). Notice the House Edge on Banker (in bold):

 Wager  Permutations  Probability  Break-Even House Edge
 Player   2,230,518,282,592,260   0.4462466093                                             1.02768                                          0.012351
 Banker   2,292,252,566,437,890   0.4585974226                                             0.97307                                          0.010579
 Tie      475,627,426,473,216              0.09516                                             9.50906                                          0.143596
 Total   4,998,398,275,503,360

The House Edge means, theoretically in the long-term, for every dollar wagered, a player is expected to lose 1.05 cents.

This is a table with the House Edge on Baccarat, comparing what the House Edge would be if Banker were paid half on wins of 1 to 9:

Banker Win On: Permutations Probability Banker Win on Any Other Result Player Win Difference House Edge
1         24,291,119,898,624 0.004859781                  2,267,961,446,539,260                  2,230,518,282,592,260 –   49,588,723,896,320.0 -0.009920923
2         44,681,581,871,104 0.00893918                  2,247,570,984,566,780                  2,230,518,282,592,260 –   39,393,492,910,080.0 -0.007881223
3         72,927,778,568,192 0.01459023                  2,219,324,787,869,700                  2,230,518,282,592,260 –   25,270,394,561,536.0 -0.005055698
4      163,359,790,133,248 0.032682428                  2,128,892,776,304,640                  2,230,518,282,592,260     19,945,611,220,992.0 0.003990401
5      216,715,928,915,968 0.043357075                  2,075,536,637,521,920                  2,230,518,282,592,260     46,623,680,612,352.0 0.009327724
6      269,232,304,455,680 0.053863716                  2,023,020,261,982,210                  2,230,518,282,592,260     72,881,868,382,208.0 0.014581045
7      384,279,324,919,808 0.076880493                  1,907,973,241,518,080                  2,230,518,282,592,260   130,405,378,614,272.0 0.026089433
8      529,914,458,673,152 0.106016854                  1,762,338,107,764,740                  2,230,518,282,592,260   203,222,945,490,944.0 0.040657614
9      586,850,279,002,112 0.117407667                  1,705,402,287,435,780                  2,230,518,282,592,260   231,690,855,655,424.0 0.04635302

Notice that on Banker wins on 1 – 3, if the House were to pay half, there would be a long-term negative edge for the House.

For wins on 4 and 5, the edge is below that of normal Baccarat. At wins on 6, that is where the House Edge comes closest to that of normal Baccarat.

If the House were to pay half on a Banker win that best profited the House, then the obvious choice would be to pay half on Banker wins on 9, which gives a House Edge 4 times that of normal Baccarat.

Do realize, that these numbers are based on an 8-deck shoe, before any cards have been dealt. Once cards are removed from the shoe, the probabilities change dynamically.

Good luck.

Video gaming and life

In the past months, I have had ample time to sample some of the many forms of video games.

Most games feature some form of simulation of real life, from racing to squad combat. AND for some of us, we spend way too much time in these environments.

The real reason, which we may not realise at first, is the sense of accomplishment it provides. That extra level up, that unique weapon or that opponent defeated; video games offer a sense of gratification for effort expended.

So great is this feeling that we look forward to the end of our work days so we can get back to gaming.

Video games have a lot to teach us in the corporate world, in terms of how we look at work, our ways of recognising and rewarding effort and even the structures of our corporate environment.

Video games always contain inherent traits, much of which can be applied to the real world:

  1. Measurable and surmountable challenges

All quest-type video games present players with tasks.  An example would be to collect a certain amount of items.  What if the game told players to collect items, but did not specify the quantity?  Sounds unlikely?  All too likely in the real world!

Create an environment where objectives are clear, have tangible rewards and are measurable. Staff know when the goals you have set are attainable and they will strive for them.
Remember that only things that get measured, get done.
If one gives unclear, ambiguous or unattainable goals, everyone knows that you do not reward or promote on any other basis than your own opinion.  This is simply based on the fact that there is no way to tell if an objective has been completed.

That, my friend, is a dictatorship or worse, idiocy.

2. Tangible and effective rewards (be it a substantial improvement in gaming performance or even just aesthetics)

When quests do get completed, gamers are rewarded with a new item or a power-up.  This can be extremely incentivizing as it not only marks achievement, but enables players to achieve higher goals.

If you are thinking that the reward system in games are instantaneous and thus inapplicable, you’d be wrong.  Players are known to perform repetitive tasks or ‘grind’ for days to accomplish a goal.  All because they know that there is a point to it.

Too often, that is not the case in many working environments.  Not because there is a shortage of repetitive tasks, but that there is a sore lack of purpose.

Tangible

Reward staff in a substantial way.  The key here is to maintain a sustainable reward system that keeps people going for the goal.

Some industries have achieved this by creating a multi-layered hierarchy of senior, assistant, deputy and vice positions. While this does create a sense of achievement, it can hamper the operations of your organization because you have just created a tall reporting structure -which is not dynamic and is ill-suited to responding to change.

An alternative would be to use lateral forms of rewards such as time-off, incentive trips or even nominal pay raises.

Effective

Reward staff in a way that motivates them to do better.  A good way is to provide special developmental programs or even higher level duties.  There is nothing that says ‘trust’ more than the opportunity to do important work.

It is important that you don’t use this as a way to get subordinates to do YOUR work.  People can smell a lazy-ass boss who is passing off work that he doesn’t want to do.  Don’t be that guy.

3. Fair treatment

The controlled environment of a game world lets regular players know when someone is cheating.  Tell-tale signs like an impossible amount of level ups is a sure sign of cheating and for the most part, the gaming community and administrators lay the smack down pretty efficiently.

Unfortunately, this does not apply to real life and it is common to see the unscrupulous gain the upper hand in a corporate environment.

Fair treatment is important as it motivates your best guys to continue giving their best as they know there is a point to it all.

If too many of the clowns get rewarded, the rest will know that the game is rigged and will stop trying or worse.  This creates a toxic environment of back-biting, politics and misery.  Don’t do that!

4. In Closing…

Promote and reward only the exemplars in your organization.  Remember that people are watching who you promote and will emulate their behavior.

Paying attention to these factors can make the difference between a team of highly motivated professionals and a set of feet-draggers that would like nothing better than to see the organization fail.

Isn’t it time we apply these principles to how we work?

 

 

 

Casino Gaming: Theoretical Win, Expected Value and Standard Deviation

Theoretical win and the standard deviations of expected value have been widely covered in numerous articles.  Unfortunately for me, the discussion on the topic has seldom been comprehensive enough to be understood by the layperson (meaning me).

This post attempts to explain the issue in the simplest possible way.

Why is this important?

Theoretical win is derived from the probabilities built into any casino game.  As all casino games are designed (in theory) to guarantee a return to the casino, the theoretical win (winnings for the player) is always negative while the expected value, also known as expectation (winnings for the casino) is always positive.

However, like with all probabilities, an element of randomness exists.  The standard deviation of theoretical win thus provides a threshold for casino managers to decide if play has passed the limit where it becomes suspect.

Theoretical Win and Expected Value

The average of results is determined by the theoretical win formula.  Hence, 50% of all results will achieve this result.

For example:

A split wager on single-zero roulette pays 17:1.

The probability of winning a split bet is 2/37 (2 numbers wagered on from a total of 37 numbers)

The probability of losing the bet is 37-2 / 37 = 35/37 (for all the other numbers not wagered on)

The theoretical win formula is:

(Probability of winning x payout in terms of units of wager) – (Probability of losing x wager)

(2/37 x 17) – (35/37 x 1) = -0.02703

This means that 50% of all play on split wagers in single-zero roulette will (in the long-term) achieve a loss of $0.02703 per dollar wagered by the player.

Standard Deviation

The majority of results will fall within 3 standard deviations from the average.

bellcurve

(Source: http://schools-wikipedia.org/)

In order to obtain the standard deviation, we turn to the binomial distribution, which is a distribution where only 1 of 2 outcomes is possible.  In our case, the outcomes being a win or a loss.

The formula for a binomial standard deviation is:

Square Root (number of games x probability of winning x probability of losing) x average wager x (unit of wager + payout in terms of units of wager)

Using our current split wager example, with an average wager of $2 over 3 games, the formula would be as follows:

Square Root(3 x 2/37 x 35/37) x $2 x (1+17)

=Square Root(3 x 2/37 x 35/37) x $2 x 18

= 0.391659 x $2 x 18

= $14.09972 (standard deviation of theoretical win)

The Result

To find out if the win of a player is improbable, we use the Z-table and the following formula:

Win – Average Win (Theoretical Win)

Standard Deviation

Our example will yield the following result should the player have won $15.  The player played 3 games with an average wager of $2, so the total wager would be $6.

15 – (-0.02703 x $6)

14.09972

=1.075352

The result is 1.075352, meaning that the result of the player’s gaming is 1.075352 standard deviations from the average.  This means that the player’s gaming results are within the top 15% of all results.  Cause for some concern.

There you have it.  Good luck!

www.onlinecasino.us