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Combinatorics, Conditional Probability and Beyond] Consider a game with n E N participants which are somehow ordered (P₁, P2, Pn). The game starts with the first numbered player (P₁) Cosses a fair coin until the first "Tail" appears. By x₁ we denote the number of flips made by P₁. Player P₁ 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 P₁ wins (express the result as a function of y). ...

Combinatorics, Conditional Probability and Beyond] Consider a game with n E N participants which are somehow ordered (P₁, P2, Pn). The game starts with the first numbered player (P₁) Cosses a fair coin until the first "Tail" appears. By x₁ we denote the number of flips made by P₁. Player P₁ 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 P₁ wins (express the result as a function of y). ...

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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[Combinatorics, Conditional Probability and Beyond] Consider a game with n E 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 P₁. Player P₁ 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 2 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 P₁ 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 P₁). 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 P₁ and P2, and obtain the expected number of winners in this version of the game. Compare results with those found in (d).
Transcribed Image Text:[Combinatorics, Conditional Probability and Beyond] Consider a game with n E 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 P₁. Player P₁ 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 2 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 P₁ 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 P₁). 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 P₁ 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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