I give Solution for the lazy =(35/36)^24=50.86% so that the prob of winning for Chevalier de Méré is 1-50.86%=49.44% so the game is negatively edged against him and not worth. Whereas if it was 25 instead of 24 he would have a positive edge but as I said above there was no calculator at that time so his error is excusable . History said that he proposed this game after he made a fortune by proposing a more simple one which was to roll 4 dices and to bet on at least one six: his probability of losing is then (5/6)^4=48.23% so he has a real edge here 50%-48.23%=1.77% Very near the Casino roulette edge when betting on Black or Red. Now like Casino, Chevalier de Méré was rich enough - he was a noble man - to benefit from this small edge and it was necessary that the edge is small so that his game partners would attribute his superior gain to chance because if they suspect he had a big edge they would refuse his rules : this is just psychology . So a casino or stock market need somehow to maintain or at least make believe that the game is at leat (nearly) fair for people or they won't play - except for the unconditional ones.
At least this was the former definition of probability (Laplace's definition) but "modern" probability has transformed and there are now two schools of thoughts: the frequentists (corresponding to the above definition) and the subjectivists (a bit like fundamentalists and technical analysts in stock market ) see slideslow here : http://www.utas.edu.au/docs/humsoc/philosophy/ccc/slides/6b.html Chance Coincidence and Chaos Laplace First Principle The first of these principles is the definition itself of probability, which, as has been seen, is the ratio of the number of favorable cases to that of all the cases possible. Second Principle. But that supposes the various cases equally possible. If they are not so, we will determine first their respective possibilities, whose exact appreciation is one of the most delicate points of the theory of chance. Then the probability will be the sum of the possibilities of each favorable case. 1814, A Philosophical Essay on Probabilities The subjectivist theory Frank Ramsey (1903-1930) Probabilities are a measure of degrees of rational partial belief.
When you say "few walk with their winnings" do you mean just for the day or forever ? . Because if they don't go away forever, it doesn't change much as each party of black jack is independant from each other probabilistically speaking. I also remark that people now use the term "money management" to avoid the proper term that is martingale mathematically because when there is no edge you can do whatever you want it won't change it. Gamblers seem to be very difficult to be convinced by that whereas it is rigourously mathematically demonstrated. Doubling at each lost is a martingale, but reducing at each lost also, although it is commonly called "anti-martingale" rule or more nobly "money management" - to make things appear more serious - it doesn't change the mathematical fact that it is still a martingale. Why do some people still think that such rule works (I restrict the question to Casino game not to stock market) ? Because statistically speaking they have applied it and it may have workedby chance and if they really win big then they have passed a level of fortune where their risk of ruin is lowered ... except if they continue to apply the same martingale over and over again with the same proportion as before then you will see the same people relose everything. This is sure for gambling as the edge is unconstestably known enough, whereas in stock market it is not as well established because the probability is much more subjective see thread above about the frequentist and subjectivist definition of probability. In the case of a casino game the frequentist approach is suitable. In the case of stock market no. In the case of Casino a martingale or anti-martingale rule cannot work except by chance, in stock market the answer is more controversial.
I use the same definition as you. For even money bets in European roulette, p = 19/37, q = 18/37, so the edge is equal to p - q = 1/37 = 2.70% (and not 1.35%).
Actually...study up on the "Gambler's Ruin" theorem in any statistics book. A casino beats the player not because its odds favor the casino, although this does help, but because it has more money than you do. In fact, even if the odds were 70% in the player's favor, the player would eventually go bust due to "unlucky streaks" that will make him run out of capital. Conversely, if the casino did not set table limits, and was faced with a player who had unlimited capital. The player could merely double his bet every time he lost, and eventually beat the house in an unlimited string of victories of his base wager. I'd be happy to explain this mathematically if you so desire.
I stand by all I said and see no confusion. I also agree with all your comments (except for the above sentence).
OK if you want . Just remark than the term edge doesn't belong to prob vocabulary but to common vocabulary. So if you want to define the edge as the gain expectancy E(G) you can, in fact it's a good definition. But in the context of the question I asked it is an "overinflated" answer as the question was much more simpler "is it worth or not ?" than the question to which you answered "what is the EXACT expected gain ?". Since I'm lazy I prefer to make one operation to answer the question than 2 operations: p=1-q and then E(G)=p-q or of course E(G)=2*p-1 but still it requires more operations than p-0.5
Every bet on the roulette table has the same casino edge.All payoffs are bases on a "fair" table of 36 numbers. The addition of 1 zero(European) or 2 zeros(American) provides the "edge". Las Vegas - 2.7% London- 5.26%