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Computer Science 174 - Fall 1998 - Sinclair - Midterm 1

# Computer Science 174 - Fall 1998 - Sinclair - Midterm 1 -...

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CS 174, Midterm #1, Fall 1998> CS 174, Fall 1998 Midterm 1 Professor A. Sinclair Read these instructions carefully 1. This is a closed book exam. Caculators are permitted 2. This midterm consists of 10 questions. The first seven questions are multiple choice: the remaining three require written answers. 3. Answer the multiple choice questions by circling the correct answer. You should be able to answer all of these from memory, by inspection, or with a very small calculation. Incorrect answers may attract a negative score, so if you do not know the answer you should not guess. 4. Write you answers to the other questions in the spaces provided below them. None of these questions requires a long answer, so you should have enough space; if not, continue on the back of the page and state clearly that you have done so. Show all your working. 5. The questions vary in difficulty: if you get stuck on some part of a question, leave it and go on to the next one. Problem 1. Two standard decks of 52 cards are randomly shuffled separately. The expected number of cards that are at the same position in both decks is 1/52 1/2 1 2 13 26 Problem 2. Three fair six-sided dice are rolled. Given that at least one of the dice comes up 6, the probability that exactly one of them comes up 6 is 25/216 75/216 125/216 25/91 75/91 1/3 Problem 3. Alice and Bob are two people in a group of size n. The group is ordered randomly in a line. The probability that there are exactly k people between Alice and Bob is file:///C|/Documents%20and%20Settings/Jason%20Rafte. ..%20Fall%201998%20-%20Sinclair%20-%20Midterm%201.htm (1 of 5)1/27/2007 6:46:39 PM

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CS 174, Midterm #1, Fall 1998> k/n (n-k)/(n*(n-1)) (n-k-1)/n! (n-k-1)/ (n*(n-1)) 2(n-k-1)/(n*(n-1) Problem 4. Each cereal box contains one coupon, chosen independently and uniformly at random from a set of n different coupons. (a) The expected number of boxes that need to be bought before a copy of some particular coupon is obtained is n^2 n*(ln (ln n)) n^(1/2) n n*(ln n) e^n (b) The expected number of boxes that need to be bought before at least one copy of all
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Computer Science 174 - Fall 1998 - Sinclair - Midterm 1 -...

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