Exercise 5. You play a two player game. You play as player 1. The two players take it in tu to draw balls out of an urn. (a) In the first version of the game, there are 4 balls in the urn. Three are white, and on gold. On their turn, each player randomly draws one ball out of the urn (the ball is not replac If the ball is white, the game continues, but if it is gold, then the player who drew it wins the ga (i) Calculate the probability that Player 1 wins the game on their first turn. (ii) Calculate the probability that Player 1 wins the game on their second turn. (iii) What is the probability that Player 1 wins the game?
Exercise 5. You play a two player game. You play as player 1. The two players take it in tu to draw balls out of an urn. (a) In the first version of the game, there are 4 balls in the urn. Three are white, and on gold. On their turn, each player randomly draws one ball out of the urn (the ball is not replac If the ball is white, the game continues, but if it is gold, then the player who drew it wins the ga (i) Calculate the probability that Player 1 wins the game on their first turn. (ii) Calculate the probability that Player 1 wins the game on their second turn. (iii) What is the probability that Player 1 wins the game?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Exercise 5.** You play a two-player game. You play as player 1. The two players take it in turns to draw balls out of an urn.
(a) In the first version of the game, there are 4 balls in the urn. Three are white, and one is gold. On their turn, each player randomly draws one ball out of the urn (the ball is not replaced). If the ball is white, the game continues, but if it is gold, then the player who drew it wins the game.
(i) Calculate the probability that Player 1 wins the game on their first turn.
(ii) Calculate the probability that Player 1 wins the game on their second turn.
(iii) What is the probability that Player 1 wins the game?
(b) In the second version of the game, 2 of the 4 balls are white, 1 is gold and 1 is black. If on their turn, the player draws the gold ball, then, as before, they win, but if they draw the black ball they lose (i.e. the other player wins).
Show using conditional probability calculations that the probability that Player 1 wins the game is 1/2.
(c) The games in (a) and (b) can be played with any number of balls.
(i) For the game in (i) suppose that we play with n balls, n - 1 of which are white and 1 of which is gold. Explain why Player 1 has an advantage (is more likely to win than Player 2) if n is odd, but there is no advantage to either player if n is even.
(ii)* [Optional] Can you explain why no player has an advantage in the game from (b). Start with n = 4. Can you again explain in words why no player has any advantage? Would you expect your answer to change if we used a different number, n of balls, assuming that n - 2 are white, 1 is gold and 1 is black? You can try the calculation for a few different values of n to develop intuition for your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb47fe1c9-1465-464b-a626-f1fe4168d136%2F368ee409-1d74-4083-8f16-20aac62ec99d%2Fkmnykr_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 5.** You play a two-player game. You play as player 1. The two players take it in turns to draw balls out of an urn.
(a) In the first version of the game, there are 4 balls in the urn. Three are white, and one is gold. On their turn, each player randomly draws one ball out of the urn (the ball is not replaced). If the ball is white, the game continues, but if it is gold, then the player who drew it wins the game.
(i) Calculate the probability that Player 1 wins the game on their first turn.
(ii) Calculate the probability that Player 1 wins the game on their second turn.
(iii) What is the probability that Player 1 wins the game?
(b) In the second version of the game, 2 of the 4 balls are white, 1 is gold and 1 is black. If on their turn, the player draws the gold ball, then, as before, they win, but if they draw the black ball they lose (i.e. the other player wins).
Show using conditional probability calculations that the probability that Player 1 wins the game is 1/2.
(c) The games in (a) and (b) can be played with any number of balls.
(i) For the game in (i) suppose that we play with n balls, n - 1 of which are white and 1 of which is gold. Explain why Player 1 has an advantage (is more likely to win than Player 2) if n is odd, but there is no advantage to either player if n is even.
(ii)* [Optional] Can you explain why no player has an advantage in the game from (b). Start with n = 4. Can you again explain in words why no player has any advantage? Would you expect your answer to change if we used a different number, n of balls, assuming that n - 2 are white, 1 is gold and 1 is black? You can try the calculation for a few different values of n to develop intuition for your answer.
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