O D D S $mart Gamblers have a basic understanding of the mathematical concepts related to gambling. This tutorial covers probability, statistical expectation, casino percentage and standard deviation. It also describes the $mart Gambler program report and how to use the information it provides. PROBABILITY The theory of probability is as follows. If an event is certain to happen its probability is 1.00. If an event is impossible to happen its probability is 0.00. Gambling events have a probability of winning and a probability of losing. The sum of those probabilities is 1.00. The term odds is commonly used to describe the probability of a gambling event winning. Consider a coin toss. There is a 0.50 probability that it would end up heads and a 0.50 probability that it would end up tails. The sum of the probabilities of a coin toss is: (0.50 heads) + (0.50 tails) = 1.00 This could be also called having 50/50 odds of one side ending up over the other side. Consider a casino roulette wheel. There are 38 numbers, 1 to 36 plus 0 and 00. The probability or odds of the number 16 coming up is one out of 38. The probability for winning and losing is: (1/38ths for the 16) + (37/38ths for the others) = 1.0000 The probabilities expressed decimally is: (.0263 for the 16) + (.9737 for the others) = 1.0000 STATISTICAL EXPECTATION When the winning and losing amounts are factored with the probabilities, the statistical expectation of a gambling event is developed. This will provide a picture of what the gambler can expect to happen over the long run. To get the statistical expectation, first multiply the gain amount by the probability of winning. Next add to it the loss amount multiplied by the probability of losing. The loss amount is expressed as a minus number. If $1 was bet on heads coming up in the coin toss, the statistical expectation would be: ($1.00 win times 0.50) + (-$1.00 times 0.50) = 0.00 This would be called a fair odds game because the expectation is neither plus or minus. It is also called having true odds. In the long run a gambler would expect to end up even if they bet on either heads or tails always. Consider what would happen if instead of $1.00 being won, $1.25 was won. The statistical expectation would then be: ($1.25 win times 0.50) + (-$1.00 times 0.50) = +0.125 Expressed as a percentage, the expectation would be +12.5%. After 100 bets, the expected result would be a gain of $12.50. This would be unfair odds. Consider what would happen if instead of $1.00 being won, $0.75 was won. The statistical expectation would then be: ($0.75 win times 0.50) + (-$1.00 times 0.50) = -0.125 Expressed as a percentage, the expectation would be -12.5%. After 100 bets, the expected result would be a loss of $12.50. This also would be unfair odds. CASINO PERCENTAGE If a casino offered gambling at fair odds they would not be able to remain in business. They would not be able to pay for buildings, utilities, employees, taxes and other expenses and show a profit. To offer fair odds and succeed they would have to be lucky in the long run. This would be gambling. If there is one thing a corporate casino owner doesn't want to do, it is to gamble. Obviously they also would not give unfair odds favoring the player. Casinos offer gambling games that either have: (1) A built-in probability favoring their winning (2) A built-in payoff schedule biasing the statistical expectation in their favor (3) A combination of these If a casino did offer a coin toss game, the payoffs would likely be similar to the unfair odds example with $0.75 paid for winning. This would give them an unfair advantage percentage - 12.5%. It is unlikely that they would offer such a game because the unfair advantage would easily be recognized by the players. They also could bias the results of the coin toss by adding a weight to one side and only taking bets on the other side. That would equally be easily recognized by the players. Casinos generally prefer games in that their advantage is not obvious to players. Consider the roulette game example provided earlier. There are 38 numbers, 1 to 36 plus 0 and 00. The probability or odds of one number coming up is 1 out of 38. However the casino pays only 35 to 1 for selecting one number and winning. The statistical expectation if $1.00 is bet on the number 16 is: ($35.00 times 1/38ths) + (-$1.00 times 37/38ths) = -0.0526 Expressed decimally: ($35.00 times .0263) + (-$1.00 times .9737) = -0.0526 (0.9211) + (- 0.9737) = -0.0526 There is 1 chance in 38 that 16 will come up and 37 chances in 38 that it won't. The -0.0526 is usually expressed as -5.26%. This is the casino percentage advantage or as it's commonly called the "PC." Sometimes it is called the "vigorish" or "vig." For every $100.00 bet on roulette, the casino can expect to retain or "hold" an average of $5.26. The casino is not gambling. There may be streaks and players may win in the short run, but over the long run they will win this percentage without fail. Time is on the side of the casino. A casino may experience a losing shift, day, week or month but over time the PC will assert itself. Casino percentage advantages range from -0.01% to -25.00% and sometimes even more. There is tremendous difference in playing a low PC game versus a high PC game. $mart Gamblers only play at the very lowest PC games available. Consider the casino's PC as the cost of the entertainment of gambling. A typical casino gambler may bet an average of $5.00 and experience an average 100 outcomes per hour. The cost of playing a -0.60% game such as six-deck blackjack would be $3.00 per hour. It is assumed that the blackjack game is played correctly as described in the Blackjack tutorial. The cost of playing a -25.00% game like the Wheel of Fortune would be $125.00 per hour. There is no particular skill required to play the Wheel of Fortune as discussed in the Other games tutorial. The difference between the two games is $122.00 per hour! These two games are available in most casinos and often right next to each other. The tutorial on Games discusses the aspects of the different casino games. Review the tutorials on the individual games to get full information on the casino percentage advantages and the best bets and other strategies. STANDARD DEVIATION Standard deviation is a measure of dispersion. It describes how far the results can vary from the expected or theoretical result. The actual calculation of it is best left to scientific calculators and computers. The following is a simplified example of standard deviation. Consider again the coin toss. The expected result would be that half the time heads would come up and half the time tails would come up. In 100 tosses, the player might expect heads to come up 50 times, an even split. Some gamblers would say that the "law of averages" will cause this to occur. In statistics, there is actually no such thing as a law of averages. While the the most frequent result will be 50 times, it may only occur about 8% of the time over a sample of 100 tosses. Random events such as a coin toss result in random results. The standard deviation for this example could be 5. In 68.3% of the times, the result will be between plus and minus one standard deviation of the expected average. For this example that would be between 45 and 55. In 95.5% of the times the result will be between plus and minus two standard deviations of the expected average. For this example that would be between 40 and 60. In 99.7% of the times the result will be between plus and minus three standard deviations of the expected average. For this example that would be between 35 and 65. This could be considered the greatest deviation likely. The streaks in random events must be respected. Consider again playing a fair odds coin toss game, betting $1 at a time and this standard deviation example. At the end of 100 events, the result could be anywhere from -$30 (won 35 times) to +30 (won 65 times). This would be a plus or minus 30% result. As the number of events increase, the percentage away from the expected result will become less. However, the absolute fluctuation amount will become greater. At the end of 10,000 events, the result could be expected to be closer to an even split percentage wise but could be substantially away in the absolute amount. The standard deviation for this example would be 50. At the end of 10,000 events the result could be anywhere from -300 (won 4850 times) to +$300 (won 5150 times). This would be a plus or minus 3% result. This is the law of large numbers, a statistical term. The longer period a gambler plays, the closer they will be to the expected result percentage wise. However they will require a greater bankroll to with stand the absolute fluctuations in negative streaks. These examples were based on a fair odds game. The impact of the standard deviation is greater on unfair odds games as offered by casinos. Effectively the plus and minus range is shifted accordingly to the casino's advantage or PC. Games with a small PC are shifted very little. Games with a large PC are shifted a great deal. Casinos have two advantages besides their built-in percentages. First they have large numbers of gamblers making bets over extended periods of time. That reduces the percentage from the expected result. Second they have huge bankrolls to handle the absolute fluctuations that may occur in a negative streak. $MART GAMBLER REPORT The $mart Gambler program report shows statistical items that can be meaningful to monitoring gambling and improving results. Of course the value of the report is directly related to the quality of the data inputted. In the Planning menu, a data collection form is available to record gambling results in the casino. Some players may prefer a small notebook but in either case it is essential to record the data as soon as practical after playing. On a side note, casinos can be funny about players keeping records. Most don't care but some may. In general it is best to be a little discrete when recording data. Use the Report-Input menu selection to input new data into the $mart Gambler program as soon as practical. Use the Report-Edit menu selection to review the data and correct any mistakes. Use the Report-Compute menu selection to update the report. If there is no data available at this time, it is recommended to select File-Example to understand better the following report items. A sample database and report will be loaded. Then either select Report-View or Report-Print. Summary statistics show the following items: Number of Sessions: The number of entries inputted. Number Win: The quantity with an even or plus result. Percent Win: (Number of Sessions / Number Win) * 100 Cum Plus/Minus: Cumulative summation of individual sessions Change since last report: Difference of the last cum to the new Best Session: The maximum individual session win Average Session: The arithmetic mean average of the sessions Worst Session: The maximum individual session loss +3 Std Deviations: The average + 3 standard deviations +2 Std Deviations: The average + 2 standard deviations Standard Deviation: The measure of dispersion from the average -2 Std Deviations: The average - 2 standard deviations -3 Std Deviations: The average - 3 standard deviations The plus and minus three standard deviations show what can be expected in 99.7% of the times. The plus and minus two standard deviations show what can be expected in 95.5% of the times. Detail statistics show the following items: Results by Casino Code: Code: Codes as set up in Planning and inputted in Report-Input Casino: Explanation of codes as set up in Planning-CasinoCodes Number of Sessions: The number of entries with that code Cum Plus/Minus: Cumulative summation of sessions with that code Average Session: The average of the sessions with that code Std Dev Session: The dispersion of sessions with that code Results by Type Code: Code: Codes as set up in Planning and inputted in Report-Input Type: Explanation of codes as set up in Planning-TypeCodes Number of Sessions: The number of entries with that code Cum Plus/Minus: Cumulative summation of sessions with that code Average Session: The average of the sessions with that code Std Dev Session: The dispersion of sessions with that code The $mart Gambler report provides many benefits. By getting the data in control, it leads to greater control over gambling. This will not reduce the entertainment aspect of gambling but increase it for a specific budget. By managing and controlling their gambling, a player can remain in the game or "action" for a longer period. The summary statistics show over all results with the cum plus/minus the most meaningful item. The average and standard deviation calculations show what is being experienced statistically. If the results differ significantly from the expected results, a review of the game played and control in the casino may be beneficial. The report also transforms what is "thought" to what is "real." A player may think they like to play at a particular casino because they think they do well there but the actual statistical results may not agree. Like wise a player may think they do well at a particular type of game but the actual statistical results may differ. The report shows the casinos and types of games that the gambler is most successful at playing. The results by casino code and type code are sorted by the average session to make it easier to comprehend and identify the best ones. In addition to the report, by selecting Report-Graph, graphs on the cum plus/minus, casino and type code average session are available. SUMMARY The $mart Gambler report helps the player understand and appreciate the mathematics of casino gambling. It also helps in making decisions on where to play and the types of games to play. The data when analyzed is a valuable tool in improving success in casino gambling. Copyright 1992 PC Information Systems All rights Reserved