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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016Distinct Symbols in a Palindromic Access CodeA 5-character access code is formed by choosing each character independently and uniformly from \ A, B, C, D, E, F\ (repetition allowed). The code is then accepted only if it is a palindrome (reads the same forwards and backwards). Among all palindromic codes, find P( at least 3 distinct symbols appear in the code ).概率中等数值题未尝试免费022Divisibility Duel with Split DrawsIn a two-stage experiment, you first draw 2 numbers without replacement from \ 1, 2, 3, 4, 5, 6\ , then independently draw 1 number uniformly from \ 1, 2, 3, 4, 5\ . Let P be the product of all three drawn numbers. Find P(6 \mid P), the probability that the product is divisible by 6.概率中等数值题未尝试免费023Tournament Bracket with Seating ConstraintsIn a round-robin chess mini-tournament, 5 players are randomly assigned to 5 boards (one player per board, all 5! assignments equally likely). The boards are numbered 1 through 5 in a circle, so board 6 wraps to board 1. Player i has a "comfort zone" consisting of boards i and i + 1 (cyclically: player 5's comfort zone is \ 5, 1\ ). The organizer wants every player to be outside their comfort zone. Find P( no player is assigned to a board in their comfort zone ).概率困难数值题未尝试面试订阅072Independent Events Covering the Sample SpaceLet A and B be independent events with P(A \cup B) = 1. (a) Using the independence condition and inclusion-exclusion, prove that (1 - P(A))(1 - P(B)) = 0. (b) What does this force about P(A) and P(B)? (c) On \Omega = \ 1,2,3,4\ with uniform probability, find events A and B that are independent, satisfy P(A \cup B) = 1, and have P(A) = 1 while P(B) = 1/2. Verify all three conditions.概率简单derivation未尝试免费075Fixed Points of a Random Permutation Are Not IndependentA permutation of \ 1,2,3,4\ is chosen uniformly at random (each of the 4! = 24 permutations equally likely). For i = 1,2,3,4, define the event A i = \ (i) = i\ (element i is a fixed point). (a) By counting, show that P(A i) = 1/4 for every i, and P(A i \cap A j) = 1/12 for every pair i j. (b) Are A i and A j independent? (c) Compute P(A i \cap A j \cap A k) for distinct i,j,k and P(A 1 \cap A 2 \cap A 3 \cap A 4). (d) Despite the failure of pairwise independence, verify the classical inclusion-exclusion identity: P\bigl(\bigcup i=1 4 A i\bigr) = 1 - 1 2! + 1 3! - 1 4! .概率困难derivation未尝试免费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128Maximum of Three Dice Equals FourYou roll three fair six-sided dice. What is the probability that the maximum of the three values equals exactly 4?概率中等数值题未尝试免费140Three-of-a-Kind Among Five DiceYou roll 5 fair six-sided dice. What is the probability that at least one face value appears exactly 3 times?概率困难数值题未尝试免费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215Distribution of Dice Sums via Probability Generating FunctionsLet X 1, X 2, \ldots, X n be iid rolls of a fair d-sided die, so each X i is uniform on \ 1, 2, \ldots, d\ . Let S n = X 1 + X 2 + \cdots + X n. (a) Derive the PGF G X 1 (s) = E[s X 1 ] in closed form. (b) Write the PGF of S n and use it to derive E[S n] and Var (S n). (c) For n = 3 fair six-sided dice (d = 6), use the PGF to find P(S 3 = 10). (d) Explain how the coefficient-extraction approach relates to the classical stars-and-bars counting with inclusion-exclusion for this problem.概率困难derivation未尝试免费219Distribution of the Maximum of Independent Geometric Random VariablesLet X 1, X 2, \ldots, X n be independent Geometric (p) random variables with P(X i = k) = (1-p) k-1 p for k = 1, 2, \ldots Define M = \max(X 1, \ldots, X n). (a) Show that P(M \le m) = [1 - (1-p) m] n for m = 1, 2, \ldots (b) Derive P(M = m) from the CDF. (c) Express E[M] as an infinite series using the tail-sum formula E[M] = \sum m=0 P(M > m). Simplify to: E[M] = \sum m=0 [1 - (1 - (1-p) m) n ]. (d) For the special case n = 2, p = 1/2, compute P(M = 1), P(M = 2), P(M = 3) and verify they sum to nearly 1. Compute E[M] exactly by evaluating the series. (e) For general n and small p, argue heuristically that E[M] \approx \ln n p by comparing to the continuous exponential analogue.概率困难derivation未尝试免费701Three-Set Region Recovery 1In a universe of 140 objects, sets A, B, C satisfy |A|=62, |B|=55, |C|=49, |A∩B|=22, |A∩C|=18, |B∩C|=17, and |A∩B∩C|=9. How many objects lie in exactly one of the three sets? How many lie in exactly two? How many lie in none?脑筋急转弯简单数值题未尝试免费706All-Letter String Count 1How many length-7 strings over an alphabet of size 3 use every letter at least once?脑筋急转弯简单数值题未尝试免费711All-Server Assignment Count 1How many assignments of 6 labeled jobs to 3 labeled servers use every server at least once?脑筋急转弯简单数值题未尝试免费716Filtered Divisibility Count 1Among the integers from 1 to 180, how many are divisible by none of 2, 3, or 5?脑筋急转弯简单数值题未尝试免费