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Unformatted text preview: Probability with Applications—Spring 2008 Problem Set 3 for ISYE 2027, Section B (March 27, 2008) Purpose. The third exam will focus on material largely covered in Chapters 5 and 6 of Walpole et al. However, this should be considered a cumulative exam in that prior material will be presupposed and may be directly treated. In the lecture periods prior to the exam, I will solve the practice exam at the board, along with additional exercises as time permits. Please see also the document “Study Guide 3,” which provides a theoretical overview of the new material you are responsible for. Also important will be your lecture notes and the document “Homework 3 Solutions.” Notes, books, and electronic devices may not be used during the exam. Practice exam. The following expressions are provided without context, as a memory aid: ( n x ) p x (1 p ) n x n ! x 1 ! ··· x k ! p x 1 1 ··· p x k k ( k x )( N k n x ) ( N n ) ( x 1 k 1 ) p k (1 p ) x k e λ ( λ x x ! ) 1 √ 2 πσ e ( x μ ) 2 / 2 σ 2 λe λx 1. Suppose we are to repeatedly toss a biased coin where the probability of seeing heads is 2 3 . Suppose that each toss is independent of any collection of other tosses. (a) What is the expected value of the number of tosses required in order to see heads come up once ? Give a simplified fraction. (b) What is the probability that the number of tosses required in order to see heads come up twice is exactly 4? Give a simplified fraction. 2. An urn contains 6 balls: 2 are white and 4 are black. (a) If we randomly draw 3 balls with replacement, what is the probability that exactly two of the balls we draw are white? Give a simplified fraction. (b) If we randomly draw 3 balls without replacement, what is the probability that exactly two of the balls we draw are white? Give a simplified fraction. 3. An urn contains 6 balls: 2 are white and 4 are black. (a) If we randomly draw 3 balls with replacement, what is the expected number of white balls drawn? Give a simplified fraction....
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 Spring '08
 Zahrn
 Normal Distribution, Poisson Distribution, Probability theory, Binomial distribution, Exponential distribution

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