2.We play a game with a deck of 52 regular playing cards, of which 26 are red and 26are black. They're randomly shuffled and placed face down on a table. You have the optionof "taking" or “skipping" the top card. If you skip the top card, then that card is revealedand we continue playing with the remaining deck. If you take the top card, then the gameends; you win if the card you took was revealed to be black, and you lose if it was red. If weget to a point where there is only one card left in the deck, you must take it. Prove that youhave no better strategy than to take the top card – which means your probability of winningis 1/2.Hint: Prove by induction the more general claim that for a randomly shuffled deck of n cardsthat 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.

Given a game with a deck of 52 regular playing cards, of which 26 are red and 26 are black.

Answered: 2. We play a game with a deck of 52…

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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Suppose you have a standard 52-card deck (four suits - spades, clubs, diamonds, hearts. 13 cards ofeach suit - 2 through 10, ace, jack, queen, king, where the last three types are considered face cards),and you are playing a game with another person where you each are dealt two cards. How manyways are there to do each of the following?(a) You get the same kind of cards as your opponent (e.g., you both get a 2 and 3, or both get aqueen and king, or both get two aces, etc.). (b) All four cards between the two of you are from different suits. (c) You are both dealt two cards of the same suit, but the suit you have is different from youropponent (e.g., your opponent has two spades, but you have two diamonds).
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You are given a deck of n cards, which are numbered 1 through n. After the n cards are randomly shuffled, the cards are dealt face up on the table, one card at a time. Card rule: after the first card is placed on the table, each new card must have a higher number than the previous card. If it does, this new card remains on the table. If the new card is lower in value, then this card is removed from the table and the game is immediately over. ***The image is the example*** Questions: Prove that for all positive integers n ≥ 3, if there are n cards in the deck, you score exactly 1 point with probability 12 and exactly 2 points with probability 13.

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