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097Bertrand's Paradox: The Random Chord ProblemConsider a circle of radius r with an inscribed equilateral triangle. A chord is drawn 'at random.' What is the probability that the chord is longer than the side of the triangle? Compute the answer under each of the following three methods of selecting a random chord: (a) **Random endpoints:** Fix one endpoint on the circle and choose the other endpoint uniformly on the circumference. (b) **Random midpoint:** Choose the midpoint of the chord uniformly in the disk. (c) **Random radius:** Choose a radius, then choose a point uniformly along that radius as the midpoint of the chord. For each method, set up and evaluate the relevant integral or geometric argument.概率简单derivation未尝试免费098The Will Rogers Paradox (Stage Migration)Group A consists of the values \ 1, 2, 3, 9\ and Group B consists of \ 5, 6, 7, 8\ . (a) Compute the mean of each group. (b) Now move the element 5 from Group B to Group A. Recompute the mean of each group. Verify that both means have increased. (c) The overall mean across all 8 values is unchanged. Prove the following general statement: if an element is moved from group B to group A, and the element's value x satisfies A < x < B (where A , B are the pre-move group means), then both post-move group means strictly exceed their pre-move values. (d) A medical study reports that a new diagnostic technology has improved average survival time in both Stage I and Stage II cancer patients, yet overall survival is unchanged. Explain how this is possible without any actual improvement in treatment.概率中等数值题未尝试免费101Matching Suits in a Two-Card DrawTwo cards are drawn without replacement from a standard 52-card deck. What is the probability that both cards are of the same suit?概率简单数值题未尝试免费102First Ace vs. First KingA standard 52-card deck is shuffled uniformly at random. What is the probability that the first ace appears before the first king when dealing cards from the top?概率简单数值题未尝试免费103Probability of a Full HouseA 5-card poker hand is dealt from a standard 52-card deck. What is the probability of being dealt a full house (three cards of one rank and two cards of another rank)?概率中等数值题未尝试免费104Probability of a FlushA 5-card poker hand is dealt from a standard 52-card deck. What is the probability that all five cards are of the same suit (a flush, including straight flushes)?概率中等数值题未尝试免费105Expected Number of Suits in a Poker HandA 5-card hand is dealt from a standard 52-card deck. Let S be the number of distinct suits represented in the hand. Using indicator random variables, find E[S].概率困难derivation未尝试面试订阅106At Least One King in a Poker HandA 5-card hand is dealt from a standard 52-card deck. What is the probability that the hand contains at least one king?概率简单数值题未尝试免费107Probability of Exactly One PairA 5-card poker hand is dealt from a standard 52-card deck. What is the probability that the hand contains exactly one pair (two cards of one rank and three cards of three different other ranks, with no other matching ranks)?概率中等数值题未尝试免费108Adjacent Cards in a Shuffled DeckA standard 52-card deck is shuffled uniformly at random and laid out in a row. What is the probability that the ace of spades and the king of spades are adjacent (next to each other)?概率中等数值题未尝试免费109Expected Wait for the First HeartA standard 52-card deck is shuffled uniformly at random. Cards are turned over one at a time from the top. Let X be the position of the first heart. Find E[X].概率中等derivation未尝试免费110The 4-3-3-3 Bridge Hand DistributionA 13-card bridge hand is dealt from a standard 52-card deck. What is the probability that the hand has a 4-3-3-3 suit distribution (exactly four cards of one suit and exactly three cards of each of the other three suits)?概率困难derivation未尝试面试订阅111Three Cards, Three Different SuitsThree cards are drawn without replacement from a standard 52-card deck. What is the probability that all three cards are from different suits?概率简单数值题未尝试免费112Drawing a Pair of AcesTwo cards are drawn without replacement from a standard 52-card deck. What is the probability that both cards are aces?概率简单数值题未尝试免费113Probability of a Poker StraightA 5-card hand is dealt from a standard 52-card deck. What is the probability that the hand is a straight but not a straight flush (i.e., five cards of consecutive ranks, not all of the same suit)? Aces can be high or low (A-2-3-4-5 through 10-J-Q-K-A).概率中等数值题未尝试免费114Expected Cards to See All Four SuitsA standard 52-card deck is shuffled uniformly at random and cards are turned over one at a time from the top. Let X be the number of cards turned over until all four suits have appeared at least once. Find E[X].概率困难derivation未尝试面试订阅115Void in a Bridge HandA 13-card bridge hand is dealt from a standard 52-card deck. What is the probability that the hand contains a void — that is, at least one suit is completely absent from the hand?概率困难derivation未尝试面试订阅116Both Top Cards RedA standard 52-card deck is shuffled uniformly at random. What is the probability that the top two cards are both red?概率简单数值题未尝试免费117Face Card Then AceTwo cards are drawn sequentially without replacement from a standard 52-card deck. What is the probability that the first card is a face card (J, Q, or K) and the second card is an ace?概率简单数值题未尝试免费118Exactly Two Suits in a Poker HandA 5-card hand is dealt from a standard 52-card deck. What is the probability that exactly two suits are represented in the hand?概率中等数值题未尝试免费