Probability and Coinless Retail Change

In a previous post, I suggested that coins could be entirely eliminated in making retail change by making change with a single additional dollar on a probabilistic basis. This needs to be read before understanding the following problem/puzzle.

Of no particular practicality, consider the following mechanical probability generator :

Assume a transparent drum mounted and rotatable on a central horizontal axis. Further assume that it is internally divided into a top and bottom half.

Fill the top half of the drum with 99 white marbles and 1 black marble.

By pushing a button on the side of the drum, a single marble is allowed to fall from the top half of the drum to the bottom half. A mechanical counter accumulates and displays the number of button pushes since all the marbles were in the top half. A separate switch releases all the marbles in the top half and allows them to all fall into the bottom half and resets the counter to zero.

The purpose of the probability generator is to determine if an additional dollar should be given out in change, with the probability set equal to the number of cents in the indicated change due. See the previous post for details.

The problem is to determine the statistics of the minimum number of button pushes required to decide whether to distribute a dollar or not.

I'm too lazy to actually produce an answer, assuming that every amount of cents change between 1 and 99 cents is equally probable, but someone may find it interesting.

A simpler, but still non-trivial problem is to identify all of the conditions that allow the button pushing to be stopped with a dollar decision now available. We want no more button pushing than necessary.

I think that there is enough information above to see how a decision can be made, but someone may have an alternative procedure, so I'll leave it somewhat vague.

When can I stop pushing the button?

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