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171Expected Pair Count With One Holiday Twice as LikelyOn a 366-point calendar, one special holiday has probability 2/366 and each of the other 364 days has probability 1/366. For n=30 independent birthdays, what is the expected number of unordered matching pairs?概率简单数值题未尝试免费172Poisson Approximation Threshold for Near CollisionsUse the pair-Poisson approximation for birthdays on a 365-day circular calendar where a pair counts as a collision if the birthdays are the same day or one day apart. What is the smallest n for which the approximated collision probability exceeds 50%?概率简单数值题未尝试免费173Expected Number of Singleton BirthdaysFor n independent uniform birthdays on a 365-day calendar, what is the expected number of people whose birthday is unique in the sample?概率中等derivation未尝试面试订阅174Expected Number of Days With Exactly Two BirthdaysFor n independent uniform birthdays on a 365-day calendar, what is the expected number of calendar days that receive exactly two birthdays?概率中等derivation未尝试面试订阅175Expected Number of Days With at Least Three BirthdaysFor n independent uniform birthdays on a 365-day calendar, what is the expected number of calendar days that receive at least three birthdays?概率困难derivation未尝试面试订阅176Server A Empty but Server B ActiveSix labeled jobs are independently routed to four labeled servers A, B, C, D with equal probability. How many assignments leave server A empty but give server B at least one job?概率简单数值题未尝试免费177Odd Load on a Tagged ServerFive labeled packets are independently assigned to three labeled servers. In how many assignments does server A receive an odd number of packets?概率简单数值题未尝试免费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182Expected Occupied Servers After a Tagged AssignmentA tagged job is known to have been routed to server A. Then five additional labeled jobs are independently routed uniformly to the four servers A, B, C, D. What is the expected total number of occupied servers?概率中等数值题未尝试免费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188Capped Weak CompositionsSeven indistinguishable jobs are split across four labeled servers. How many occupancy vectors are possible if no server is allowed to receive more than three jobs?概率中等数值题未尝试免费190Expected Number of Odd-Load BinsEight labeled balls are independently assigned to five labeled bins. What is the expected number of bins that end up with an odd load?概率困难数值题未尝试免费191Two Tagged Balls Stay IsolatedBalls 1 and 2 are tagged. Six labeled balls are independently assigned to four labeled bins. What is the probability that ball 1 and ball 2 land in different bins and each of those two bins contains no other ball?概率简单数值题未尝试免费192Exact Occupancy Profile 2-2-1-0Five labeled tasks are independently assigned to four labeled queues. How many assignments produce queue loads that, after sorting, equal (2,2,1,0)?概率中等数值题未尝试免费194Variance of the Number of Occupied UrnsFour distinguishable balls are thrown independently and uniformly at random into 3 distinguishable urns. Let N be the number of nonempty urns. Find Var (N). Give an exact fraction.概率困难数值题未尝试免费196Strictly Descending Loads Across Three Named BinsSeven labeled packets are independently assigned to three labeled bins A, B, C. How many assignments produce loads satisfying load(A) > load(B) > load(C)?概率简单数值题未尝试免费197Expected Number of Heavy BinsEight labeled balls are independently assigned to four labeled bins. What is the expected number of bins whose load is at least three?概率简单数值题未尝试免费199Expected Maximum Urn OccupancyFour distinguishable balls are thrown independently and uniformly at random into 3 distinguishable urns. Let M = \max(X 1, X 2, X 3) be the maximum number of balls in any single urn. Find E[M]. Give an exact fraction.概率中等数值题未尝试免费200Full Distribution of Empty Urns via Stirling NumbersSix distinguishable balls are thrown independently and uniformly at random into 5 distinguishable urns. Let E denote the number of empty urns. Derive the probability mass function P(E = k) for every possible value of k, expressing each probability as an exact fraction.概率困难derivation未尝试免费204Negative Binomial Variance from Geometric SummationLet X be the number of independent Bernoulli(p) trials needed to accumulate r successes. Express X as a sum of independent geometric random variables and use this to find Var (X).概率中等derivation未尝试免费