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2. We play a game with a deck of 52 regular playing cards, of which 26 are red and 26 are black. They're randomly shuffled and placed face down on a table. You have the option of "taking" or "skipping" the top card. If you skip the top card, then that card is revealed and we continue playing with the remaining deck. If you take the top card, then the game ends; you win if the card you took was revealed to be black, and you lose if it was red. If we get to a point where there is only one card left in the deck, you must take it. Prove that you have no better strategy than to take the top card – which means your probability of winning is 1/2.

2. We play a game with a deck of 52 regular playing cards, of which 26 are red and 26 are black. They're randomly shuffled and placed face down on a table. You have the option of "taking" or "skipping" the top card. If you skip the top card, then that card is revealed and we continue playing with the remaining deck. If you take the top card, then the game ends; you win if the card you took was revealed to be black, and you lose if it was red. If we get to a point where there is only one card left in the deck, you must take it. Prove that you have no better strategy than to take the top card – which means your probability of winning is 1/2.

College Algebra
College Algebra
10th Edition
ISBN:9781337282291
Author:Ron Larson
Publisher:Ron Larson
Chapter8: Sequences, Series,and Probability
Section8.6: Counting Principles
Problem 74E: Lottery Powerball is a lottery game that is operated by the Multi-State Lottery Association and is...
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2. We play a game with a deck of 52 regular playing cards, of which 26 are red and 26 are black. They're randomly shuffled and placed face down on a table. You have the option of "taking" or “skipping" the top card. If you skip the top card, then that card is revealed and we continue playing with the remaining deck. If you take the top card, then the game ends; you win if the card you took was revealed to be black, and you lose if it was red. If we get to a point where there is only one card left in the deck, you must take it. Prove that you have no better strategy than to take the top card – which means your probability of winning is 1/2. Hint: Prove by induction the more general claim that for a randomly shuffled deck of n cards that are red or black - not necessarily with the same number of red cards and black cards - there is no better strategy than taking the top card.
Transcribed Image Text:2. We play a game with a deck of 52 regular playing cards, of which 26 are red and 26 are black. They're randomly shuffled and placed face down on a table. You have the option of "taking" or “skipping" the top card. If you skip the top card, then that card is revealed and we continue playing with the remaining deck. If you take the top card, then the game ends; you win if the card you took was revealed to be black, and you lose if it was red. If we get to a point where there is only one card left in the deck, you must take it. Prove that you have no better strategy than to take the top card – which means your probability of winning is 1/2. Hint: Prove by induction the more general claim that for a randomly shuffled deck of n cards that are red or black - not necessarily with the same number of red cards and black cards - there is no better strategy than taking the top card.
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