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I am thinking a problem of drawing a ball in a sealed box. Assuming there is a box, contains plenty red and white balls, the number of red and white balls are unknown but let's assume there will be ##p## chance to draw a red ball and ##q=1-p## chance to get a white one. Those probability is constant throughout the calculation. Let assume that at beginning we start the game by picking one ball out of the box, if it is a red one, there will be 3 more draws given; otherwise, game over.

For each draw out of 3, if you get a red ball, 3 more draws added other wise, use up one chance until all draws used up. I wonder if there is any way to estimate the maximum draws could be at the given ##p## and ##q##?

I can estimate the average draws to be about 3.01993 if #p=0.006659#. But how to find the maximum possible number of draws? I write a program to simulate this process for billions times and I see that in some game I could get up to 21 draws. Is it any way to compute this number in theory? Thanks.

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# I Finding the average time with given probability

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