Pascal discovered that the chevalier wasn’t giving himself an even chance, let alone a favorable one.  His mathematical procedure was fairly sophisticated  (specifically, he multiplied the odds against card winning by the colog of the hyperbolic log of two, i.e., 0.693, in case 35 x 0.693=  24.255); but the equation proved that to give himself an even chance, de Méré would have to throw the diece 24,255 times.  Of course, you can’t throw anything a fractional number of times.  But had he bet on throwing a double Six once in every 25 throws, he would have had slightly better than an even chance of winning.

This slight edge is usually called a “favorable percentage.”  In the chevalier’s case, the favorable percentage was 0.745 per cent (arrived at by subtracting 24.255 from 25).  In other words, the odds in favor of his throwing a double Six in 25 throws were 0.745 per cent better than even.

Favorable percentages are extremely important to online poker gamblers who operate a “house” (that is, who put themselves in the position of enduring a long run by accepting bets from all comers).  They must give themselves a slightly favorable percentage in order to remain in business against virtually unlimited time and capital.  For however big the house’s capital is to begin with, and however long it remains continually in business, it can never match the aggregate time and capital available against it.

To illustrate how a gambling house adjusts its  odds, here is a hypothetical case in terms of roulette:
Imagine a roulette wheel excluding the zero and the American double zero – that is, with only 36 spaces.  And imagine that you have $ 36 to bet with, and that the house is paying the true odds of 35 to 1.  You backgammon the number of your choice with a dollar at every spin.  Assume that you lose 35 times and that with your last dollar your number comes up.  Because the odds are 35 to 1, you win $ 35 plus the dollar you bet with on that spin.  You have in fact broken even and are backgammon where you started with $ 36.  But this would never do for a casino, which has to make a profit as well as stay in business.  It cannot make a profit solely from losers, since in the long run the amount of money received from losers will tend to even up with the amount paid to winners.  But if a gambling house pays out at less than the true odds to every winner, then it will gain an artificial advantage.

If the house odds in that imaginary roulette game 30 to 1 instead of the true odds of 35 to 1, you would win only $ 30.  With your dollar bet returned, you would still be $ 5 lighter in pocket than when you started.  This is the margin of profit that keeps the house in business.  If you were luckier, and your chosen number came up in the first spin of the wheel rather than the 36th, you would have $ 30 more than you started with.  But the house would still be saving itself money; it has in effect “charged”

you $ 5 for gambling privileges (the extra $ 5 that you would have won if the odds were true).  Although winners rarely realize it, it is they perhaps even more than the losers who make casinos (and book-making ) such profitable businesses.

The percentage worked out by a gambling house in its own favor varies from house to house (and with different games).  Some of them take very high percentages indeed; others, like the Monte Carlo casino, rub along on a maximum of 2 26/37 (2.7) per cent, which according to their calculations is the lowest possible figure that allows them to cover the expenses of running the place and to make a reasonable profit  without driving gamblers away.  And there is nothing fraudulent about these percentage rake-offs; most gambling centers make it clear what percentage they take, so the gamblers know (or should know) what odds are being offered before they place their bets.

In games combining skill and chance, like most card games, the most skillful player is in the same position as a casino.  His extra skill is equivalent to the artificial advantage of a favorable percentage.  The more skilled he is compare with the other players, the more quickly he will begin to win and keep his winning lead.  Chance , of course, determines the cards each player is dealt, but unless the players are equally skilled (in which case they will in the long run all break even) chance is counteracted by skill, and the less skillful players are eventually certain to lose.
Right, a 36-space roulette table (without the zeros).  Imagine a player betting a dollar a spin (on, say, the 4) who loses 35 times but wins on the36th spin.  If the casino paid at true odds of 35 to 1.  he would break even, winning $ 35 plus his last $ 1 bet.  To stay in business, casinos must pay less than true odds.  The chart below shows what would happen to that player if (with $ 60 capital) he played night similar rounds-losing 35 bets and winning the 36th at a casino playing 30 to 1.  He would lose $ 35 and win $ 31 each round; after eight rounds he would have lost his entire capital.

Innumerable fallacies have been perpetuated about chance and probability, many of which more or echo Napoleon’s remark that security is “the mathematical elimination of chance” as silly a thing as has been said by him or anybody else, since it is impossible to eliminate a natural law.  Perhaps his fallacious reasoning is responsible for the notion, staunchly clung to by soldiers, that it is mathematically impossible for two shells to fall on the same spot.  The odds against a shell’s falling on a designated spot on a battlefield are, of course, related to the aim of the gun and the size of the target area.  But they are not increased by one jot after the fall of the first shell.