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# [Combinatorics, Conditional Probability and Beyond] Consider a game with n = N participants which are somehow ordered (P1, P2, ..., Pn). The game starts

## [Combinatorics, Conditional Probability and Beyond] Consider a game with n = N participants which are somehow ordered (P1, P2, ..., Pn). The game starts with the first numbered player (P1) tosses a fair coin until the first "Tail" appears. By x1 we denote the number of flips made by P. Player P is eliminated from the game if x1 < y. In this case, the second player tosses a fair coin until the first "Tail" appears. Similarly, by x2 we denote the number of flips made by P2. Player P2 is eliminated from the game if x1 + x2 < y. The game proceeds this way until either the total number of coin flips attains y, or all players are eliminated. All those players who are not eliminated at the end of the game are winners. [This game is an abstract formalization of "crossing the glass bridge" from the popular TV show "Squid Game"]. (a) Calculate the probability that player P wins (express the result as a function of y). (b) Calculate the probability that player P2 wins (express the result as a function of y). (c) Find the expected number of tosses for a player (Ex). Only for this question, assume there is no limit on the total number of flips (y). (d) Assume that n = 2 and y = 5. Obtain numeric results for the questions from (a) and (b). (e) Suppose n = 2 and the rules of the game are slightly changed. Now players toss a coin one after the other (starting with P). A player is eliminated if Tail appears on its move and the total number of flips made by both players are less than y = 5. The game stops when either the total number of flips attains 5, or both players are eliminated. Winners are those players who are not eliminated at the end of the game. Calculate the probabilities of winning for P and P2, and obtain the expected number of winners in this version of the game. Compare results with those found in (d).

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