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# Ass4 - 1 As before continue drawing balls until the ﬁrst...

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MATH 352 Assignment #4 Due: Tuesday October 30, 2007 1. In class we looked at the following problem: An urn initially contains one red and two green balls. Balls are repeatedly drawn, and each time a ball is drawn, it is replaced into the urn along with another ball of the same colour. Balls are drawn until the red ball has been selected. We deﬁned X to be the total number of draws. We then calculated the probability mass function of X and found E ( X ) = . This may seem counterintuitive: if we decide in advance to draw n balls from the urn, the probability of drawing the red ball is p (1) + p (2) + ··· p ( n ), which converges to 1 as n → ∞ . Alternatively, P ( X n ) 0 as n → ∞ . (a) Suppose that the urn originally contains k - 1 red balls and k green balls, k 2. The example done is class is the case where k = 2. Is it true that E ( X ) = for all values of k ? (b) Now suppose the urn originally contains twice as many greens balls as red, ie. the urn contains k red and 2 k green balls, k
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Unformatted text preview: 1. As before, continue drawing balls until the ﬁrst red is drawn. However, this time, each time a ball is drawn it is returned to the urn along with k more of the same colour. The problem done in class corresponds to the case where k = 1. Is it true that E ( X ) = ∞ for all values of k ? 2. Chapter 4 Theoretical Exercise 31 (29 in 6th edition). 3. The probability of throwing a 6 with a biased die is p , where 0 < p < 1. Three players A , B , and C take turns rolling this die, starting with A , then B , then C . The ﬁrst one to throw a 6 wins. (a) Find the probability of winning for players A , B , and C . (b) If X is a random variable which takes the value r if the game ﬁnishes with the r th throw, ﬁnd the probability mass function for X , and evaluate E ( X ) and V ar ( X ). 4. Chapter 4 Theoretical Exercise 20. 5. Chapter 4 Theoretical Exercise 28 (26 in 6th edition)....
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