## 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.