Answer

In both his calculations, the gambler used the simplified addition rule. This overlooks the fact that the outcomes he is dealing with are not mutually exclusive; he may get a 6 (or a double 6) in more than one roll. To calculate the true probabilities, we need to use the full addition rule.

In fact, the simplest way to proceed is to calculate the probability of getting no 6 in four throws. Since the throws are independent, we can use the simplified multiplication rule, which tells us that the probability of getting no 6 in four throws is = 0.482. Since the probability of getting at least one 6 covers all the other outcomes, we can use the subtraction rule to calculate this probability; it is 10.482 = 0.518.

Similarly, we can calculate the probability of getting no double 6 in 24 throws. Since the probability of not getting a double 6 in one throw is 35/36, and the throws are independent, the probability of getting no double 6 in 24 throws is $(35/36)^{24}$ = 0.509. Again, by the subtraction rule, the probability of getting at least one double 6 is 10.509 = 0.491. This is lower than the probability of getting at least one 6 in four throws, as the gambler had noticed.