## Concepts: The Math Behind The Lotto Max

On 22 Jun 2021, Canada’s Interprovincial Lottery Corporation awarded its largest ever Lotto Max prize pool in the amount of \$140 million (\$70 million for the grand prize and 70 \$1 million MaxMillion prizes). It thus seems timely to have a look at the math behind lotteries. The website states that the odds of winning the Lotto Max are 1 in 33,294,800. This is correct to a point, but misleading. Let’s have a look at the rules of the Lotto Max: Players choose 7 numbers out of 50 Numbers cannot be repeated Numbers are automatically sorted into ascending order Each play buys 3 lines Each play costs \$5 Seeing that players choose 7 out of 50 numbers that are non-repeating, the equation for the total number of possible combinations (this is different from permutations where the order in which the numbers appear is significant) when playing the Lotto Max is: 50! / (7! x 43!) The ! sign is a

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## Baccarat – An Analysis on Side Wagers

There is something to be said about card counting in Baccarat. The immense number of permutations often makes it difficult to card count in this game without some form of assistance. The fact that different card values have different effects on probability in this game and that these effects are not always consistent, is another puzzling aspect. The following table shows the expectations of several side wagers using various permutations of card values with between 5 to 25 cards being drawn for each. This is done to show that the advantage can be obtained from card counting and that one does not need to go very far into the shoe for it. Note that a positive expectation here indicates that the player would have the advantage in the following coup on that wager. Now, some of these situations seem very specific and they are. This data is not meant to show that card counting is viable, but that it is

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## 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

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## 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

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## 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

## Baccarat: Conditions for Positive Expectation on Side Wagers

In simulating different conditions of a Baccarat shoe, the following instances were observed: Wager refers to the wager type studied – Pairs, Wins on Natural 9s (WN9) and Wins on Natural 8s (WN8). Win % Probability indicates the percentage probability of that result occurring. 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. 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

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## 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

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## What video games can teach companies

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:

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## 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

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## The Logic Of The Losing Shoe

The Logic of the Losing Shoe For those in the casino industry, especially for us surveillance folks, the words ‘losing shoe’ are all too familiar. A losing shoe is a period of play, normally lasting the length of one shoe of cards (which may be from one to as many decks as the shoe can hold!), which registers a substantial loss. Ever wondered how that loss limit was set? In considering this question, it is useful to once again refer to our central limit theorem.  Here is the graph again. (Source: http://schools-wikipedia.org/) The central limit theorem proposes that up to 99.9% of all occurrences happen between -3 to -1 and 1 to 3 σs from the average or mean.  σ extends into the positive (meaning 1 to 3 σ) and negative (meaning -1 to -3 σ). In order to derive any sort of boundary in a casino game, one has to calculate the following: Probability Of Events Occurring (for more

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