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003Checkpoint-Exclusive Grid RoutesA robot moves from (0,0) to (5,3) using 5 right steps and 3 up steps in random order. How many paths visit exactly one of the checkpoints A=(2,1) and B=(4,2)?概率中等数值题未尝试免费005Status Strings With Exactly One Flat StepHow many length-6 strings over the alphabet L, M, H use all three symbols and have exactly one adjacent equal pair?概率困难derivation未尝试面试订阅008Three-State Paths With Exactly Two SwitchesA 5-day signal path is recorded using the symbols B, S, H . How many such paths have exactly two day-to-day switches and end in a different state from where they started?概率中等数值题未尝试免费009Reverse-Engineering Overlap from Frequency CountsIn a risk system, three alert types A, B, C can fire simultaneously. From historical logs you know: - P(A) = 0.5, P(B) = 0.4, P(C) = 0.3, - P(A \cap B) = 0.2, but P(A \cap C) and P(B \cap C) are unknown, - P( none of A,B,C) = 0.1, - P( exactly one fires ) = 0.7. Determine P(A \cap B \cap C) and P( exactly two fire ). Show all steps.概率中等derivation未尝试面试订阅018Colorings With Profile 3-2-1-0 and Different EndsSix lockers are colored using the palette R, G, B, Y . Count the colorings in which one color appears 3 times, one appears 2 times, one appears once, one does not appear at all, and the first and last lockers have different colors.概率中等数值题未尝试免费019Distinct Ordered Triples With a Modular Sum and a Large EntryHow many ordered triples (a,b,c) of distinct elements chosen from 0,1,2,3,4,5,6,7,8 satisfy a+b+c ≡ 1 (mod 3) and max(a,b,c) > 5?概率中等数值题未尝试免费020No Fixed Points Among the First ThreeHow many permutations of 1,2,...,8 have no fixed point among positions 1, 2, and 3?概率困难数值题未尝试面试订阅021Four-Subsets With Exactly One Consecutive PairHow many 4-element subsets of 1,2,...,10 contain exactly one consecutive pair?概率中等数值题未尝试免费052Pairwise but Not Mutual IndependenceLet the sample space be \Omega = \ 1, 2, 3, 4\ with uniform probability P(\ i\ ) = 1/4. Define A = \ 1, 2\ , B = \ 1, 3\ , C = \ 1, 4\ . Show that A, B, C are pairwise independent but not mutually independent.概率简单derivation未尝试免费053Zero Correlation Does Not Imply IndependenceLet X be uniformly distributed on \ -1, 0, 1\ . Define Y = X 2. Compute Cov (X, Y) and determine whether X and Y are independent.概率中等derivation未尝试免费054Conditional Independence Breaks Under MarginalizationA coin is chosen at random: with probability 1/2 it is fair (p = 1/2) and with probability 1/2 it is biased (p = 1, always heads). Let A be the event that the first flip is heads and B the event that the second flip is heads. Show that A and B are conditionally independent given the coin type C, but that A and B are not marginally independent. Compute P(B \mid A).概率中等derivation未尝试免费055Independent Events Become Dependent Under ConditioningLet A and B be independent events with P(A) = P(B) = 1/2. Define D = A \triangle B (exactly one of A, B occurs). (a) Compute P(D), P(A \mid D), and P(A \cap B \mid D). (b) Are A and B conditionally independent given D? (c) Are A and B conditionally independent given D c?概率困难derivation未尝试面试订阅058XOR Pairwise Independence Without Mutual IndependenceLet X and Y be independent Bernoulli (1/2) random variables. Define Z = X \oplus Y (where \oplus denotes addition modulo 2). (a) Show that Z \sim Bernoulli (1/2). (b) Show that any two of \ X, Y, Z\ are independent. (c) Are X, Y, Z mutually independent? Exhibit a specific joint event that violates the mutual independence condition.概率中等derivation未尝试免费059All Triples Independent but Quadruple NotLet \Omega = \ 0, 1, 2, \ldots, 7\ with uniform probability P(\ \omega\ ) = 1/8. Write each \omega in binary as (b 2, b 1, b 0). Define events: A = \ \omega : b 0 = 1\ , \quad B = \ \omega : b 1 = 1\ , \quad C = \ \omega : b 2 = 1\ , \quad D = \ \omega : b 0 \oplus b 1 \oplus b 2 = 1\ . (a) Show that A, B, C are mutually independent. (b) Show that any triple chosen from \ A,B,C,D\ is mutually independent. (c) Show that A, B, C, D are not mutually independent.概率困难derivation未尝试面试订阅060Independence from Boolean Combinations Requires Mutual IndependenceLet A, B, C be events. (a) Prove that if A, B, C are mutually independent, then A is independent of B \cap C c. (b) Now let \Omega = \ 1,2,3,4\ with uniform probability, A = \ 1,2\ , B = \ 1,3\ , C = \ 1,4\ . Verify that A, B, C are pairwise independent but not mutually independent. (c) Compute P(A \cap (B \cap C c)) and P(A) P(B \cap C c). Does the independence A \perp\!\!\perp (B \cap C c) hold?概率困难derivation未尝试面试订阅064Independence Is Not Closed Under UnionsLet A, B, C be events with A \perp\!\!\perp B and A \perp\!\!\perp C. (a) Prove that if A, B, C are mutually independent, then A \perp\!\!\perp (B \cup C). (b) Now let \Omega = \ 1,2,3,4\ with uniform probability, A = \ 1,2\ , B = \ 1,3\ , C = \ 1,4\ . Verify that A \perp\!\!\perp B and A \perp\!\!\perp C. (c) Show that A and B \cup C are not independent. Why doesn't pairwise independence suffice?概率中等derivation未尝试免费065Mixtures of Independent Distributions Destroy IndependenceA biased coin lands heads with probability 1/2. If heads, set (X, Y) = (1, 1). If tails, draw X and Y independently, each Bernoulli (1/2). (a) Compute the full joint distribution of (X, Y). (b) Show that X and Y have the same marginal distribution. (c) Are X and Y independent? Prove your answer. (d) Compute P(X = 1 \mid Y = 0) and compare it to P(X = 1). Interpret the result.概率困难derivation未尝试面试订阅067Identical Marginals Do Not Imply IndependenceThree cards are labeled (0,0), (0,1), and (1,0). One card is drawn uniformly at random. Let X be the first number and Y the second number on the drawn card. (a) Compute P(X = 0), P(X = 1), P(Y = 0), and P(Y = 1). (b) Are the marginal distributions of X and Y the same? (c) Are X and Y independent? Verify by checking whether P(X = x, Y = y) = P(X = x) P(Y = y) holds for all (x, y) \in \ 0,1\ 2.概率中等derivation未尝试免费069An Event Independent of Itself Must Be TrivialLet A be an event in a probability space. (a) Write the independence condition P(A \cap A) = P(A) P(A) and deduce which values of P(A) satisfy it. (b) On \Omega = \ 1,2,3,4\ with uniform probability, verify your answer by testing A = \ 1\ , A = \ 1,2\ , A = \emptyset, and A = \Omega. (c) Interpret: what does it mean probabilistically for an event to be independent of itself?概率中等derivation未尝试免费070Pairwise and Triple Independence Without Mutual Independence of Four EventsLet \Omega consist of all binary strings of length 4 with an even number of 1s, each equally likely: \Omega = \ 0000,\, 0011,\, 0101,\, 0110,\, 1001,\, 1010,\, 1100,\, 1111\ . Define events A i = \ \omega \in \Omega : \omega i = 1\ for i = 1,2,3,4. (a) Show that P(A i) = 1/2 for each i. (b) Verify that every pair is independent: P(A i \cap A j) = 1/4 for all i j. (c) Verify that every triple is independent: P(A i \cap A j \cap A k) = 1/8 for all distinct i,j,k. (d) Compute P(A 1 \cap A 2 \cap A 3 \cap A 4) and compare with P(A 1) P(A 2) P(A 3) P(A 4). Are the four events mutually independent?概率困难derivation未尝试面试订阅