The gambler's problem

Suppose you roll a single die four times; what is the probability of rolling at least one 6? The gambler reasoned that since the chance of a 6 in each roll is 1/6, the chance of a 6 in 4 rolls is 4 $\times$ 1/6 = 2/3. Now suppose you roll a pair of dice 24 times; what is the probability of rolling at least one double 6? The gambler reasoned that since the chance of a double 6 in one roll is 1/36, the chance of a double 6 in 24 rolls is 24 $\times$ 1/36 = 2/3. In other words, the gambler expected to win each bet 2/3 of the time. His problem was that he seemed to lose more often with the second bet than the first. He was at a loss to explain this, so he asked his friend Pascal for an answer.