Model a process as a poisson process

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I just answered this question on coloring balls at a fixed rate - Interesting Original Probability Question. This problem seems to have all the ingredients to model as a poisson distribution with rate = 1 ball per minute. But the expected value calculation is off by a large value. Can we model this problem as a poisson distribution? More generally, if a process has a fixed rate of events (1 ball per minute) that happen independently and randomly, then I can apply poisson process. Are there any other conditions?