An event either will happen or will not happen; this constitutes a certainty. Some events are dependent, others independent. The difference is very important. Independent events have no connection, their happenings neither forwarding nor obstructing one another. Choosing a card from each of two distinct packs includes two independent events; for the taking of a card from the first pack does not in any way affect the taking of a card from the second--the chances of drawing, or of not drawing, any particular card from the second pack being neither lessened nor increased. On the other hand, the taking of a second card from a pack from which one has already been drawn is a dependent event, as the composition of the pack has been altered by the abstraction of one particular card. The surprising way in which an apparently small advantage operates may be judged from the following example:--A and B agree to play for one guinea a game until one hundred guineas are lost or won. A possesses an advantage on each game amounting to 11 chances to 10 in his favour. Mathematical analysis of this advantage proves that B would do well to give A upwards of ninety-nine guineas to cancel the agreement. Further, many speculative events, which at first sight seem to be advantageous to one side, are demonstrated by mathematical investigation to be of an exactly contrary nature. A bets B thirty-two guineas to one that an event does not happen, and also bets B thirty guineas even that it does happen in twenty-nine trials. Besides this A gives B one thousand guineas to play in this manner six hours a day for a month. Here B would appear to have some advantage. Mathematical investigation, however, proves that in reality the advantage of A is so great that B ought not only to return the thousand guineas to A, but give him, in addition, another ten thousand guineas to cancel the agreement. Every game of chance presents two kinds of chances which are very distinct--namely, those relating to the person interested (the player) and those inherent in the combinations of the game. That is to say, there is either "good luck" or "bad luck," which at different times gives the player a "run" of good or bad fortune. But besides this, there is the chance of the combinations of the game, which are independent of the player and which are governed by the laws of probability. Theoretically, chance is able to bring into any given game all the possible combinations; but it is a curious fact that there are, nevertheless, certain limits at which it seems to stop. A proof of this is that a particular number at roulette does not turn up ten or a dozen times in succession. In reality there would be nothing astounding about such a run, but it is supposed never to have happened. On the other hand, the numbers in one column at roulette have been known not to turn up during seventeen successive coups.