Among the many mistakes that are committed about chances, one of the most common and least suspected was that which related to lotteries. Thus, supposing a lottery wherein the proportion of the blanks to the prizes was as five to one, it was very natural to conclude that, therefore, five tickets were requisite for the chance of a prize; and yet it is demonstrable that four tickets were more than sufficient for that purpose. In like manner, supposing a lottery in which the proportion of the blanks to the prize is as thirty-nine to one (as was the lottery of 1710), it may be proved that in twenty-eight tickets a prize is as likely to be taken as not, which, though it may contradict the common notions, is nevertheless grounded upon infallible demonstrations. When the Play of the Royal Oak was in use, some persons who lost considerably by it, had their losses chiefly occasioned by an argument of which they could not perceive the fallacy. The odds against any particular point of the ball were one and thirty to one, which entitled the adventurers, in case they were winners, to have thirty-two stakes returned, including their own; instead of which, as they had but twenty-eight, it was very plain that, on the single account of the disadvantage of the play, they lost one-eighth part of all the money played for. But the master of the ball maintained that they had no reason to complain, since he would undertake that any particular point of the ball should come up in two and twenty throws; of this he would offer to lay a wager, and actually laid it when required. The seeming contradiction between the odds of one and thirty to one, and twenty-two throws for any chance to come up, so perplexed the adventurers that they began to think the advantage was on their side, and so they went on playing and continued to lose.