Hw3 - x = 0(fall or x = 10(survival What is the probability that the particle eventually survives 4 Problem Consider the following symmetric random

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Homework # 3, Due Tuesday, January 31 1. Problem. At each turn, a gambler bets a certain amount, wins it with probability p and loses it with probability q = 1 - p . When he begins playing, he has $ k for some integer k > 0 and his goal is to reach $ n for some integer n > k , after which he stops playing. He also stops playing if he loses all his money. The gambler has the option of betting $1 at each turn and the option of betting $0 . 5 at each turn. Show that betting $0 . 5 is a better option if p > 1 / 2 and that betting $1 is a better option if p < 1 / 2. 2. Problem. A gambler starts playing having $10. At every step, he wins $1 with probability 1 / 2 and loses $1 with probability 1 / 2. What is the probability that after 50 turns he ends up with $20 and never goes below $5 in the process? 3. Problem. Let us consider the following random walk. A particle starts at x = 5 and with probability 1 / 2 stays in place, with probability 1 / 3 moves one step to the right and with probability 1 / 6 moves one step to the left. The walk stops when the particle reaches
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Unformatted text preview: x = 0 (fall) or x = 10 (survival). What is the probability that the particle eventually survives? 4. Problem. Consider the following symmetric random walk with no barriers. A particle starts at x = 0. With probability 1 / 2 the particle moves one step to the right and with probability 1 / 2 the particle moves one step to the left. Prove that the probability that the particle returns to x = 0 for the Frst time after 2 n steps is 1 2 n-1 p 2 n n P 2-2 n . 5. Problem. a) We toss a fair coin n times. Let a n be the number of outcomes which do not contain a sequence . . .HTH . . . and for which the Frst toss lands H and let b n be the number of outcomes which do not contain a sequence . . .HTH . . . and for which the Frst toss lands T . ±ind a recurrence formula relating numbers a n and b n . b) We toss a fair coin 10 times. What is the probability that there will be a sequence . . .HTH . . . ? 1...
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This note was uploaded on 02/06/2012 for the course STAT 525 taught by Professor Alexanderbarvinok during the Winter '12 term at University of Michigan-Dearborn.

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