Probability of Matching All Types of Prizes for the First Time Is Maximized at a Surprisingly Early Number of Trials

Suppose that when money or coins are placed in a box containing an arbitrary number of prizes of several different types, one of each type of prize will appear alone each time. Is the probability that all types of prizes will be together for the first time maximized when the majority of prizes are removed from the box? Simulations based on probability theory show that this probability reaches its maximum value in a surprisingly small number of trials, contrary to expectations. This will help us understand not only mathematical phenomena, but also real-world phenomena. Phenomena that do not occur without several substances or conditions seem unlikely to occur, but the results of this study suggest that, contrary to expectations, they are surprisingly likely to occur probabilistically.