Probabilities of An Infinite Sequence of Die Rolling
Problem 745
Consider an infinite series of events of rolling a fair six-sided die. Assume that each event is independent of each other. For each of the below, determine its probability.
(1) At least one die lands on the face 5 in the first $n$ rolls.
(2) Exactly $k$ dice land on the face 5 in the first $n \geq k$ rolls.
(3) Every die roll results in the face 5.
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