4802
Fall Math 2007
Probability Problems Due Tuesday, September 11: 1. Shue 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 rst 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 i 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 ght to the death. What fraction of duels lead casualties? to 4. A drunken man standing one step from the edge of a cli takes a sequence of random steps either away from or toward the cli. At any step, his probability of taking a step away from the cli is p (and therefore his probability of taking a step toward the cli is 1 p). What is the probability that the man will eventually fall o the cli? 5. If (0, 1) is irrational, prove that there is a nite game played with a fair coin such that the probability of one player winning the game is . (A game is nite if with probability one it must end in a nite number of moves.) 6. A sequence of numbers i [0, 1] is chosen at random with respect to the uniform distribution. Find the expected value of the unique integer n for which
n1 n
i 1 and
i=1 i=1
i > 1 .
1