Luck or Probability?

In everyday life, when it comes to things like gambling or buying lotteries, luck, or God's will is often thought to be the factor that determines the outcome. But if we take a mathematic perspective and consider it in light of probability, things may seem much less mysterious. In my hometown (Xiamen, China), we play a game to celebrate the Mid-Autumn Festival. A family or a group of friends, by turns, roll a set of six dice to win moon cakes. The game comes to an end when a whole set of sixty-three moon cakes of different sizes have run out. In this paper, I will discuss the probability of winning moon cakes of different sizes, and the order in which moon cakes of different sizes run out.

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So the order in which moon cakes run out is: the 4 ones, the 16 ones, the 32 ones, the 2 ones, the 8 ones, and finally, the biggest one. The reason why the biggest one is the last one to be won will be shown in the next paragraph.

It is noticeable that with case 3, as many as 200 rolls are needed to win up the eight moon cakes, while in all other cases except case 5, only eighty rolls or so are needed. One might think that this game is not that reasonable or scientific after all. But maybe the designer of the game deliberately made it this way to prevent the game from finishing too soon. My hometown folks have long noticed this and there is a saying "When there is still one of "the eight" moon cakes left, the winning of the biggest moon cake is not final." That is, before the whole set of sixty-three moon cakes