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Unformatted text preview: Math 4802 Fall 2007 Probability Problems Due Tuesday, September 11: 1. Shuﬄe an ordinary deck of 52 playing cards, and then turn the cards face up one at a time. On average, how many cards are required to produce the ﬁrst ace? 2. Show that the probability that December 25th falls on a Wednesday is not 1/7. (Leap years have 366 days; other years have 365 days. Year n is a leap year iﬀ either (a) 4 divides n and 100 does not divide n, or (b) 400 divides n.) 3. Duels in the town of Putnamville are usually not fatal. Each contestant in a duel arrives at a random moment between 6AM and 7AM on the prescribed day, stays for 5 minutes, and then leaves, unless his opponent arrives within that 5-minute interval, in which case they ﬁght to the death. What fraction of duels lead to casualties? 4. A drunken man standing one step from the edge of a cliﬀ takes a sequence of random steps either away from or toward the cliﬀ. At any step, his probability of taking a step away from the cliﬀ is p (and therefore his probability of taking a step toward the cliﬀ is 1 − p). What is the probability that the man will eventually fall oﬀ the cliﬀ? 5. If α ∈ (0, 1) is irrational, prove that there is a ﬁnite game played with a fair coin such that the probability of one player winning the game is α. (A game is ﬁnite if with probability one it must end in a ﬁnite number of moves.) 6. A sequence of numbers αi ∈ [0, 1] is chosen at random with respect to the uniform distribution. Find the expected value of the unique integer n for which
n−1 n αi ≤ 1 and
i=1 i=1 αi > 1 . 1 ...
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This note was uploaded on 03/13/2010 for the course MATH 4801 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
- Spring '08