Final answer:
The probability of rolling (4,4), (5,4), or (3,4) with two indistinguishable dice is 1/7. For distinguishable dice, the probability of the extended corresponding event is 5/36.
Step-by-step explanation:
If two indistinguishable dice are rolled, the probability of the event {(4,4), (5,4), (3,4)} can be calculated by recognizing that with indistinguishable dice, the outcomes (5,4) and (4,5) for example, are considered the same. As there are a total of 6 outcomes for each die, and since they are indistinguishable, we count only the distinct pairwise outcomes, which results in 21 possible outcomes when two dice are thrown (1,1; 1,2; ..., 6,6, but without repeating for reverse ordering, like 1,2 and 2,1). Our event has 3 favorable outcomes, giving us a probability of 3/21 = 1/7.
For a pair of distinguishable dice, the corresponding event is {(4,4), (5,4), (3,4), (4,5), (4,3)} because the order in which the dice fall matters. With distinguishable dice, we have 6 outcomes for the first die multiplied by 6 outcomes for the second die, giving us a total of 36 outcomes. There are 5 favorable outcomes in this case, so the probability is 5/36.