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131A_1_midterm06

# 131A_1_midterm06 - bets on all the positions divisible by 3...

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EE 131A Midterm Exam (25 pts; 60 min.) Winter 2006 K. Yao 1. a. In how many ways can four men and four women be seated in a row if men and women must occupy alternate seats? (2 pts) b. How many ways can they be seated if only a man is required to sit in the ﬁrst seat? (2 pts) Express both answers as integers. 2. Suppose there are three boxes numbered 1, 2, and 3 (denoted by B 1 , B 2 , and B 3 , respectively), with diﬀerent number of colored marbles in each box as shown below in the second and third columns. No . of marbles BoxNo . Red White 1 2 3 5 5 10 5 10 5 A box is selected depending on the outcomes of the toss of two fair coins. Box 1 is selected if two heads appear, Box 2 is selected if two tails appear, otherwise Box 3 is selected. a. What are the probabilities of P ( B i ) , i = 1 , 2 , 3? (1 pt) b. Let A denote the event the marble is white. What are the conditional probabilities of P ( A | B 1 ) , i = 1 , 2 , 3? (1 pt) c. What is the probability of P ( A )? (Express your result as a fraction.) (2 pts.) d. What is the conditional probability of P ( B 1 | A )? (3 pts.) 3. A wheel used in gambling can stop in 18 equally likely positions numbered from 1 to 18. A person places
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Unformatted text preview: bets on all the positions divisible by 3. a. What is the probability that the person will win exactly 3 times in 8 attempts? (2 pts) b. What is the probability that the person will win 3 or more times in 8 attempts? (4 pts) Express both of your solutions as three signiﬁcant digit decimals. 4. Consider a discrete r.v. X with P ( X = x ) = p ( x ) = 1 / 2 n , for all positive integers n = 1 , 2 ,.... a. Write the cdf F X ( x ) ,-∞ < x < ∞ , of this r.v. (2 pts) b. Sketch F X ( x ) ,-∞ < x < ∞ , and label all points of interest. (1 pt) c. Write the pdf f X ( x ) ,-∞ < x < ∞ , of this r.v. (2 pts) d. Sketch f X ( x ) ,-∞ < x < ∞ , and label all points of interest. (1 pt) e. What is the smallest x satisfying P ( X ≤ x ) = 15 / 16? (1 pt) f. What is the largest x satisfying P ( X ≤ x ) = 15 / 16? (1 pt)...
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