The Monty Hall Problem is arguably the most famous and counter-intuitive brainteaser in the history of mathematics. On a game show, you choose between three doors; behind one is a car, while the other two conceal goats. You select door number one but, before opening it, Monty Hall–the host of the show–opens door two and shows you that there is a goat behind it. He now gives you the option of sticking with door one, or switching to door three. The problem is to determine which choice gives you the better chance of winning the car. The answer obvious to most people is entirely mistaken.
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There exists a significant body of work on determining the acquisition number of various graphs when the vertices of those graphs are each initially assigned a unit weight. We study the size of residual set of the path, star, complete, complete bipartite, cycle, and wheel graphs for variations on this initial weighting scheme, with the majority of our work focusing on the acquisition number of randomly weighted graphs. In particular, we bound the expected acquisition number of the $n$-path when n “units” of integral weight, or chips, are randomly distributed across its vertices between $0.26n$ and $0.43n$. We then use Azuma’s Lemma to prove that this expected value is tightly concentrated. Additionally, we offer a non-optimal acquisition protocol algorithm for the randomly weighted path and compute the expected size of the resultant residual set.