Problem 4: Where is Max? Max has vanished, and you assume the worst! MagusCorp Security had been lingering nearby recently, and you heard them mention things like 'illegal network access', 'bagels', 'illegal human testing', and 'no one can interfere', and you worry that Max has gotten in trouble for your recent activities. You didn't mean any harm, and now your friend may be in trouble!

If you're going to stage a daring rescue, you need to know where Max is. You assume that Max is being held on one of the floors of the building, floor 1 to floor N , but which? To beat MagusCorp, you need to think like MagusCorp.

Suppose that for each floor, there is some probability pi that MagusCorp has hidden Max on floor i. Not knowing which floor Max is on, you'll pick a floor to break into randomly, picking floor j with probability qj for each floor.

1) If the pj , qj probabilities are fixed in advance, what is the probability that you pick the right floor and save Max? (I'm looking for a formula in terms of {p1, p2, ..., pN} and {q1, q2, ..., qN}.)

2) If you knew the probabilities pi, what probabilities qj should you use to maximize your probability of saving Max? (I'm looking for formulas for q1, q2, ..., qN in terms of p1, p2, ..., pN that make the formula in 4.1 as large as possible.)

But you know that MagusCorp is clever. You have to assume they know you're coming to rescue him, and will take that into account.

3) How should MagusCorp choose the probabilities pi to minimize your best possible probability of saving Max? (I'm looking for values or expressions for p1, p2, ..., pN that - assuming BagelBot chooses as in 4.2, makes the formula in 4.1 as small as possible.)

4) Assuming MagusCorp hides Max according to the above probabilities, what is the probability you are going to be able to save Max? (I'm mostly looking for you to summarize the results of the above calculation, to determine how likely you are to capture Max if both BagelBot and MagusCorp are smart.)