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001Pinned Delegation Without the Rival PairA 3-person delegation contains a preassigned chairperson H and two additional members chosen uniformly from six others. Among those six, exactly one unordered pair is a forbidden rival pair. What is the probability the final delegation does not contain that forbidden pair?概率简单数值题未尝试免费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006Overloaded Server in a Two-Stage Routing SchemeA load balancer routes 4 independent jobs to 3 servers \ S 1, S 2, S 3\ . Server S 1 already holds one pre-assigned job. Each of the 4 new jobs is routed in two stages: first a coin with P( heads ) = \tfrac 1 2 is flipped; if heads the job goes to S 1, if tails the job is sent to S 2 or S 3 each with probability \tfrac 1 2 (i.e., P(S 2) = P(S 3) = \tfrac 1 4 ). A server is called "overloaded" if it holds 4 or more jobs (counting the pre-assigned one for S 1). Construct the sample space for the 4 routing outcomes and compute P( at least one server is overloaded ).概率中等数值题未尝试免费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011Pinned Surjection to Three DesksFive labeled tasks are assigned to three labeled desks A, B, and C. Task 1 is forced to desk A. How many assignments use all three desks at least once?概率简单数值题未尝试免费012Network of One-Way BridgesA flood-prone town is connected by one-way pedestrian bridges arranged in 3 stages. Stage 1 has 2 starting points (S 1, S 2), stage 2 has 3 relay islands (I 1, I 2, I 3), and stage 3 has 2 destinations (D 1, D 2). The bridges are: S 1 I 1, S 1 I 2, S 2 I 2, S 2 I 3; and I 1 D 1, I 2 D 1, I 2 D 2, I 3 D 2. A refugee's route is a path S i I j D k using only existing bridges. If a route is chosen uniformly at random from all valid routes, find P( the route passes through I 2 \mid the refugee arrives at D 1).概率中等数值题未尝试免费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017Product Divisibility in a Token DrawAn urn holds 6 tokens labeled 1, 2, 3, 4, 5, 6. Two tokens are drawn sequentially *with replacement* (order matters). Construct the sample space and find P( the product of the two drawn numbers is a multiple of 6 but not a multiple of 12).概率中等数值题未尝试免费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025Collision Pattern in Two-Round SamplingIn round 1, you draw 3 numbers independently and uniformly at random (with replacement) from \ 1, 2, 3, 4\ . In round 2, you draw 2 numbers independently and uniformly at random (with replacement) from \ 3, 4, 5, 6\ . The two pools overlap at \ 3, 4\ . Across all 5 draws, find P( exactly one value appears more than once, and that value appears exactly twice ). Express your answer as a reduced fraction.概率困难数值题未尝试面试订阅026Conflicting Two-Screen UpdateA loan is distressed with prior probability 0.3. Screen A flags a distressed loan with probability 0.8 and flags a healthy loan with probability 0.1. Any flagged loan then goes to screen B, which passes a distressed loan with probability 0.25 and passes a healthy loan with probability 0.70. Given that A flagged the loan and B did not pass it, what is the posterior probability the loan is distressed?概率简单数值题未尝试免费027Prior Needed to Break Even at 50-50 PosteriorA signal has likelihood ratio 5 in favor of the hypothesis H relative to not-H. What prior probability p of H makes the posterior exactly 1/2 after seeing the signal?概率简单数值题未尝试免费028Factory Posterior After One Good and One Bad ItemA part comes from factory A with prior probability 0.6 and from factory B otherwise. Factory A produces defects with probability 0.1; factory B produces defects with probability 0.4. Two parts from the same unknown factory are inspected and exactly one is defective. What is the posterior probability the parts came from factory A?概率中等数值题未尝试免费