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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120Expected Distinct Ranks in a Seven-Card HandSeven cards are drawn without replacement from a standard 52-card deck. Let R be the number of distinct ranks represented among the seven cards. Find E[R].概率困难derivation未尝试面试订阅125All Four Suits in a Seven-Card HandSeven cards are drawn without replacement from a standard 52-card deck. What is the probability that all four suits are represented among the seven cards?概率困难derivation未尝试面试订阅153Birthday Collision Probability and Exponential ApproximationFifty people are in a room. Each birthday is independent and uniform on \ 1, 2, \ldots, 365\ . (a) Write the exact probability that at least two share a birthday. (b) Derive a useful upper bound on P( all distinct ) using the inequality 1-x \le e -x and simplify. How does the approximation compare to the exact value?概率中等derivation未尝试免费154Expected Number of Birthday-Collision PairsIn a group of n people whose birthdays are independent and uniform on \ 1,\ldots,365\ , let X be the number of unordered pairs (i,j) with i < j who share a birthday. Using indicator random variables, find E[X]. Then determine the smallest n for which E[X] \ge 1.概率中等derivation未尝试免费155Variance of Birthday-Collision Pair CountContinuing from the setup of the expected collision-pair count: n people have independent uniform birthdays on \ 1,\ldots,d\ . Define X = \sum i<j 1 [B i = B j]. (a) Compute Var (X). (b) A surprising intermediate step: show that Cov ( 1 [B i = B j],\, 1 [B j = B k]) = 0 for distinct i,j,k even though the two indicators share the index j. Explain intuitively why this zero covariance holds. (c) For d = 365 and n = 28, compute Var (X) numerically and give the coefficient of variation \sigma X / E[X].概率困难derivation未尝试面试订阅157Non-Uniform Birthdays Increase Collision ProbabilitySuppose d days have birthday probabilities p 1, p 2, \ldots, p d with \sum j p j = 1 (not necessarily uniform). For n people whose birthdays are independent draws from this distribution: (a) Show that for n = 2, P( collision ) = \sum j=1 d p j 2 \ge 1 d , with equality if and only if all p j = 1 d . (b) Deduce that the uniform distribution minimizes the collision probability among all distributions on d days. Give a one-line intuitive explanation for why non-uniformity helps collisions.概率中等derivation未尝试免费159Near-Birthday Problem: Birthdays Within One DayFourteen people have birthdays chosen independently and uniformly on a circular calendar of 365 days (day 1 is adjacent to day 365). Two people have a **near-match** if their birthdays differ by at most 1 day (i.e., they land on the same day or on consecutive days). Let M be the number of unordered near-match pairs. (a) Compute E[M]. (b) Using a Poisson approximation for the probability that M \ge 1, estimate P( at least one near-match ). (c) Contrast with the standard birthday problem: for n = 14 people, what is P( at least one exact match )?概率中等数值题未尝试免费160Expected and Variance of Distinct Birthday CountAmong n people whose birthdays are independent and uniform on \ 1, \ldots, d\ , let D be the number of distinct birthdays observed. (a) Derive E[D] using indicator random variables. (b) Derive Var (D). You will need P( day j and day k both occupied ) for j \ne k. (c) For n = 100 and d = 365, compute E[D], Var (D), and the expected number of "collision people" n - D (people whose birthday coincides with at least one other person). (d) Is E[n - D] the same as the expected number of collision pairs \binom n 2 /d from the indicator-pair approach? Explain the distinction.概率困难derivation未尝试面试订阅163Expected Number of Exact Triple CollisionsFor n independent uniform birthdays on an m-day calendar, what is the expected number of unordered triples of people that share an exact birthday?概率中等derivation未尝试面试订阅165Expected Number of People Who Share Someone's BirthdayFor n independent uniform birthdays on a 365-day calendar, what is the expected number of people whose birthday is shared by at least one other person?概率困难derivation未尝试面试订阅167Expected Days Hit by Both GroupsGroup A has a people and group B has b people, with independent uniform birthdays on a 365-day calendar. What is the expected number of calendar days on which both groups have at least one birthday?概率中等derivation未尝试面试订阅169Expected Pairs Within Two DaysOn a 365-day circular calendar, what is the expected number of unordered pairs among n independent uniform birthdays whose birthdays are at circular distance at most 2 days?概率困难derivation未尝试面试订阅180Hypergeometric Moments from an UrnAn urn contains 20 balls: 8 red and 12 blue. You draw 5 balls without replacement. Let X be the number of red balls drawn. Derive E[X] and Var (X) using indicator random variables. Express each answer as an exact fraction.概率困难derivation未尝试免费185Coupon Collector: Mean and VarianceA machine dispenses one of n = 4 equally likely prize types per trial. Let T be the number of trials needed to collect all 4 types. Derive both E[T] and Var (T) by decomposing T into independent geometric phases. Express each answer as an exact fraction.概率困难derivation未尝试免费189Probability of a Heavily Loaded UrnSix distinguishable balls are thrown independently and uniformly at random into 4 distinguishable urns. What is the probability that at least one urn contains 3 or more balls? Give an exact fraction.概率困难数值题未尝试免费