There seem to be two contradictory arguments. On the one hand, each of the three marbles has an equal chance of winning, and two of them belong to Alan, so it seems that there’s a 2/3 chance that Alan will win.

On the other hand, there are four possible outcomes:

(a) both of Alan’s rolls are better than Bob’s,

(b) Alan’s first roll is better than Bob’s, but his second is worse,

(c) Alan’s first roll is worse than Bob’s, but his second is better, and

(d) both of Alan’s rolls are worse than Bob’s.

In 3 of the 4 cases, Alan wins, so it appears that his overall chance of winning is 3/4.

Which argument is correct?

Source: J. Bertrand, Calcul des Probabilités, 1889, via Eugene Northrop, Riddles in Mathematics, 1975.