Peirce writes “It is an indubitable result of the theory of probabilities that every gambler, if he continues long enough, must ultimately be ruined … if he plays long enough he will be sure some time to have such a run against him as to exhaust his entire fortune. … The same thing is true of an insurance company. Let the directors take the utmost pains … according to the doctrine of chances, the time must come, at last, when their losses will bring them to a stop.” This note shows that under the most plausible interpretation of the quoted passage Peirce is committed to a mathematical untruth.