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300Robust Common Edges of Two Random Spanning TreesLet T 1 and T 2 be two independent uniformly random spanning trees of the complete graph K n (each drawn uniformly at random from all n n-2 labeled spanning trees, independently of the other). Find the expected number of edges that belong to both T 1 and T 2. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费302Robust Matching Colors in a LineThere are n people standing in a line. Each person independently and uniformly picks one of three colors: red, green, or blue. What is the expected number of adjacent pairs (i, i+1) who chose the same color? Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单数值题未尝试免费303Robust Coincident RollsYou roll a fair k-sided die n times independently. Let M be the number of pairs (i, j) with 1 \le i < j \le n such that roll i equals roll j. Find E[M]. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等derivation未尝试免费305Robust Unique Choices and Unique NeighborsThere are n people in a line, and each independently and uniformly picks an integer from \ 1, 2, \dots, k\ . A person is called unique if no other person picked the same number. (a) Using indicator variables, find E[U], the expected number of unique people. (b) A unique neighbor pair is a pair of adjacent people (i, i+1) who are both unique. Find E[N], the expected number of unique neighbor pairs. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费315Robust Singleton Coupons After Random DrawsA collector draws m coupons independently and uniformly at random from n types. A coupon type is called a singleton if it appears exactly once among the m draws. Find the expected number of singleton types. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费322Robust Adjacent Matches in Dice RollsRoll a fair six-sided die n times independently, producing a sequence D 1, D 2, \dots, D n. An adjacent match occurs at position i (for 1 \le i \le n - 1) if D i = D i+1 . Find the expected number of adjacent matches. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率简单数值题未尝试免费323Robust Overlap of Two Random SubsetsLet S and T be two subsets of \ 1, 2, \dots, n\ , each chosen independently and uniformly at random from all \binom n k subsets of size k (where 1 \le k \le n). Find the expected size of their intersection |S \cap T|. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等derivation未尝试免费324Robust Monochromatic Edges in a Random ColoringEach vertex of the complete graph K n is independently colored red or blue, each with probability \tfrac 1 2 . An edge is monochromatic if both its endpoints have the same color. Find the expected number of monochromatic edges. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等数值题未尝试免费325Robust Comparable Pairs in Random PointsLet X 1, X 2, \dots, X n be independent and uniformly distributed on [0,1] d (the d-dimensional unit hypercube). Two points X i and X j are called comparable if one dominates the other coordinatewise, i.e., either X i \le X j in every coordinate or X j \le X i in every coordinate. Find the expected number of comparable pairs. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费551Expected Occupied Labels 110 independent packets are hashed uniformly into 8 labels. What is the expected number of labels hit at least once?概率简单数值题未尝试免费556Expected Singleton Labels 18 independent arrivals are assigned uniformly to 10 labels. What is the expected number of labels that receive exactly one arrival?概率简单数值题未尝试免费561Tagged Label Survives Unseen 1A particular tag is one of 9 equally likely outcomes on each of 11 independent draws. What is the probability that this tag is never drawn?概率简单数值题未尝试免费571All Labels Seen by Draw 5 1Draw 5 coupons independently and uniformly from 3 types. What is the probability that all 3 types have appeared at least once by time 5?概率简单数值题未尝试免费5942Collecting a Full Set of SixA vending machine dispenses one of 6 distinct toy types, each equally likely and independent across purchases. What is the expected number of purchases needed until you own at least one of every type?概率中等数值题未尝试免费5948Empty Mailboxes10 letters are placed independently and uniformly at random into 8 mailboxes. What is the expected number of mailboxes that remain empty?概率简单数值题未尝试免费5952Distinct Types Across Two PacksA booster pack contains 4 cards drawn WITHOUT replacement from a pool of 9 equally likely distinct types (so the 4 cards in a single pack are all different types). You open two packs; the two packs are independent of each other (8 cards total). What is the expected number of DISTINCT types you own?概率简单数值题未尝试免费5956How Much by the DeadlineYou will draw exactly 6 coupons, each uniform and independent over 4 types. The promotion ends after these 6 draws. What is the expected number of DISTINCT types you will have collected by the deadline?概率简单数值题未尝试免费5958Bins With Exactly Two9 balls are thrown independently and uniformly into 6 bins. What is the expected number of bins that contain EXACTLY 2 balls?概率简单数值题未尝试免费5959Coupons With BlanksEach draw is a blank with probability 1/4 (no coupon), and otherwise (probability 3/4) yields one of 3 collectible types, each equally likely. What is the expected number of draws to collect all 3 types?概率中等数值题未尝试免费5990Expected Time to the Third ArrivalOrders arrive at a matching engine as a Poisson process with rate 4 per minute. What is the expected time, in seconds, until the 3rd order arrives (measured from time 0)?概率简单数值题未尝试免费