The record of a month’s roulette playing at Monte Carlo can afford us a material for discussing the foundations of knowledge.
Karl Pearson ( 1857 – 1936 ), leading founder of modern Statistics.
At the end of the eighteenth century, in the years of the French Revolution, republican chemists, physicists and mathematicians decided to make order once for all in the jungle of national European measures by implementing everywhere the decimal standard. The French scientists endeavoured to convert even time to the new revolutionary system, but they collided with the fierce resistance of the watchmakers who required to sell all old watches in stock before conforming to the new rules. In these very years the roulette game took ground even though it was based on an awkward variant of the execrated sexagesimal system.
The number 36 is the highest score when tossing six dice. When we play with one die the six possible outcomes have the same probability, 1/6. With two dice things start to become difficult. We can get 2, or 12, by one combination only (1, 1; or 6, 6;), and the relevant probability is equal to 1/36 (1/6 x 1/6 ) for both.
As for the remaining outcomes we have to take into consideration more possible combinations. We can get 3 through: 1, 2 and 2, 1; probability = 2/36. Moreover we can get 4 through: 2, 2; 3, 1; 1, 3; probability = 3/36, and so on. Things go from bad to worse when playing with 3, 4, 5 and 6 dice. In such a situation only professional gamblers have the working knowledge for betting on the more probable numbers, i.e. the ones obtainable with more combinations.
By including in the betting range the first 5 digits the roulette made a democratic revolution in the games of chance: every number, from 1 to 36, has the same odds and all the players have the same chances to win. At roulette any number, as for instance 6, has an outcome probability equal to: 1/36 = 0.02777. Instead, in a six dice – game the number 6 ( six times 1’s ) has the utmost remote probability of outcome, equal to: 1/6 x 1/6 x 1/6 x 1/6 x 1/6 x 1/6 = 1/46,656 = 0.000021433.
This means only once, on average, out of more than 46,600 six dice tosses; whilst we can get same result within 36 rounds, on average, of roulette. We say on average since our number will not be caught necessarily within 36, or 46,656 trials. According to the Law of large numbers ( Jakob Bernoulli, 1654 – 1705 ) by increasing the number of identical trials the average number of successes comes closer and closer to the relevant probability values.
In the course of a game of chance players and casino’s entrepreneurs, the bankers, are more and more exposed to the probability of ruin, the risk of losing their capital because of adverse number sequences.
At roulette the payout of any bet on a number is 36 times what placed on it. Actually there are 38 pockets in the roulette wheel, 36 for numbers from 1 to 36, plus 2 more: one for the single zero and another for the double zero. Consequently, the true probability to catch a number is: 1/38.
In Probability Theory the Mathematical Expectation is the amount obtained by multiplying the magnitude of the prize by the probability of catching it. Thus, if at roulette we bet £1 on a number our mathematical expectation is £36 (what we can win ) x 1/38 ( probability to get it ) = £ 0.947. We have the same amount in any roulette’s game: colour, carré, dozen, etc. In such a way bankers have a profit margin equal to 2/38 = 5.3%, on average, of the overall money at stake. This means that in the course of time the capital of overall players is more and more eroded whilst casino’s entrepreneurs gain more and more. Even at roulette the public can go bust, bankers cannot.