{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw3 - x = 0(fall or x = 10(survival What is the probability...

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}