Question: Problem 2.2 (Video 1.5, 1.6, Lecture Problem) Consider the following scenario. You play a simple game with probability of winning 1/4. You play this

Problem 2.2 (Video 1.5, 1.6, Lecture Problem) Consider the following scenario. You

Problem 2.2 (Video 1.5, 1.6, Lecture Problem) Consider the following scenario. You play a simple game with probability of winning 1/4. You play this game repeatedly until your third loss, and then stop playing. Assume all games are independent. (a) What is the probability of the following specific sequence of game outcomes: Win, Lose, Win, Lose, Lose? 1 (b) How many different sequences of games are there that end after exactly 5 games? (Hint: you must lose the last game to stop. There aren't that many, so you can enumerate them.) (c) What is the probability of playing exactly 5 games? (d) Given that you play exactly 5 games, what is the probability that your first game ended in a loss? (e) Now, let's generalize this a bit. Say the probability of winning an individual game is p and that you play until your mth loss. What is the probability of playing exactly k games?

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