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132Large Sum Given Odd ProductYou roll two fair six-sided dice and are told that their product is odd. What is the probability that their sum exceeds 8?概率中等数值题未尝试免费133Exactly One Pair Among Four DiceYou roll 4 fair six-sided dice. What is the probability that exactly two of the four dice show the same value and the other two dice each show values different from each other and from the pair?概率中等数值题未尝试免费134Near-Even Split in Seven Coin FlipsYou flip 7 fair coins. What is the probability that the number of heads and the number of tails differ by at most 1?概率中等数值题未尝试免费135Conditional Third Roll Reaching TenYou roll two fair six-sided dice. If their sum is even, you roll a third die and your score is the sum of all three dice. If their sum is odd, your score is just the sum of the first two dice. What is the probability that your score is at least 10?概率困难数值题未尝试免费136Two-Dice Sum Is a Perfect SquareYou roll two fair six-sided dice. What is the probability that their sum is a perfect square?概率简单数值题未尝试免费137Exactly Two Heads in Four FlipsYou flip 4 fair coins. What is the probability that exactly 2 of them land heads?概率简单数值题未尝试免费138Exactly One Pair Among Three DiceYou roll 3 fair six-sided dice. What is the probability that exactly two of the three dice show the same value while the third die shows a different value?概率中等数值题未尝试免费139Run of Three or More Heads in Eight FlipsYou flip 8 fair coins. What is the probability that somewhere in the sequence there is a run of at least 3 consecutive heads?概率困难数值题未尝试免费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141Even Product of Two DiceYou roll two fair six-sided dice. What is the probability that their product is even?概率简单数值题未尝试免费142Exactly Two Sixes in Four DiceYou roll 4 fair six-sided dice. What is the probability that exactly 2 of the 4 dice show a six?概率中等数值题未尝试免费143No Consecutive Heads in Six FlipsYou flip 6 fair coins in sequence. What is the probability that no two consecutive coins land heads?概率中等数值题未尝试免费144All Six Faces Represented in Eight DiceYou roll 8 fair six-sided dice. What is the probability that every face value from 1 to 6 appears at least once among the 8 dice?概率困难数值题未尝试免费145No Face Appears More Than Twice in Six DiceYou roll 6 fair six-sided dice. What is the probability that no face value appears more than twice?概率困难数值题未尝试免费146Heads Beat the DieYou flip 3 fair coins and independently roll a single fair six-sided die. What is the probability that the number of heads strictly exceeds the value shown on the die?概率简单数值题未尝试免费150Red Dice Meet Blue DiceYou simultaneously roll 3 red fair six-sided dice and 3 blue fair six-sided dice. What is the probability that the sum of the red dice equals the sum of the blue dice?概率困难数值题未尝试免费153Birthday Collision Probability and Exponential ApproximationFifty people are in a room. Each birthday is independent and uniform on \ 1, 2, \ldots, 365\ . (a) Write the exact probability that at least two share a birthday. (b) Derive a useful upper bound on P( all distinct ) using the inequality 1-x \le e -x and simplify. How does the approximation compare to the exact value?概率中等derivation未尝试免费156Reverse Birthday Problem: Minimum Calendar SizeTwenty-three people have birthdays chosen independently and uniformly from \ 1, 2, \ldots, d\ . What is the smallest d such that the probability of at least one shared birthday is strictly less than 1 2 ?概率简单数值题未尝试免费157Non-Uniform Birthdays Increase Collision ProbabilitySuppose d days have birthday probabilities p 1, p 2, \ldots, p d with \sum j p j = 1 (not necessarily uniform). For n people whose birthdays are independent draws from this distribution: (a) Show that for n = 2, P( collision ) = \sum j=1 d p j 2 \ge 1 d , with equality if and only if all p j = 1 d . (b) Deduce that the uniform distribution minimizes the collision probability among all distributions on d days. Give a one-line intuitive explanation for why non-uniformity helps collisions.概率中等derivation未尝试免费158Triple Birthday Collision ThresholdIn a room of n people with birthdays uniform on \ 1,\ldots,365\ , let A be the event that at least three people share the same birthday. (a) Using a Poisson approximation (model the occupancy of each day as an independent Poisson (n/365) variable), derive an approximate formula for P(A). (b) Find the smallest n such that P(A) \ge 1 2 under this approximation.概率中等数值题未尝试免费