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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181Coupon Collector with Four TypesA cereal box contains one of 4 equally likely coupon types. You buy boxes one at a time. What is the expected number of boxes you must buy to collect all 4 types?概率简单数值题未尝试免费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183Collision Probability in the Occupancy ModelFive distinguishable balls are thrown independently and uniformly at random into 12 distinguishable urns. What is the probability that at least two balls land in the same urn? Give an exact fraction.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193Conditional Expected Occupancy of a Nonempty UrnFour distinguishable balls are thrown independently and uniformly at random into 3 distinguishable urns. Given that urn 1 is nonempty, what is the expected number of balls in urn 1? Give an exact fraction.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195Expected Time to First Collision in Six UrnsBalls are thrown one at a time, each landing independently and uniformly at random into one of 6 urns. Let T be the index of the first ball that lands in an already-occupied urn (so T \ge 2). Derive E[T] and give an exact fraction.概率困难derivation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