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Computer Science 174 - Fall 1998 - Sinclair - Final

# Computer Science 174 - Fall 1998 - Sinclair - Final -...

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<!--cs174, final examination, fall/1998--> Each of the 50 states has two senators. A committee is chosen by selecting a set of 50 senators uniformly at random. (a) The probability that no senator from California is on the committee is 1/(2^50) 98!/100! C(98, 50)/100! (50!)/C(100, 50) C(98, 50)/C (100, 50) (98/100)^50 (b) The probability that the committe contains one senator from every state is (2^50)/C(100, 50) 1/C(100, 50) (50!)/C(100, 50) (2^50)/100! 1/(100!) 50!/100! Problem #3 (a) A coin with heads probability p is tossed n times. The expected number of heads is 1/ p n /2 np np (1 - p ) n/p (b) A run in a sequence of coin tosses is a maximal subsequence of either heads or tails. (Thus, for example, the sequence HHHTTHTHH contains five runs.) The expected number of runs in n tosses of a coin with heads probability p is n /2 np np (1 - p ) 2( n - 1) p (1 - p ) + 1 ( np )/(1 - p ) Problem #4 (a) There are n bins, and balls are thrown into them in sequence, independently and uniformly at random. The process stops when any three bins are occupied. The expected number of balls thrown is file:///C|/Documents%20and%20Settings/Jason%20Rafte. ..20Fall%201998%20-%20Sinclair%20-%20Final%20Exam.htm (2 of 9)1/27/2007 6:46:38 PM
<!--cs174, final examination, fall/1998--> 3 n /( n - 3) n [(1/ n ) + 1/ ( n - 1) + 1/ ( n - 2)] C( n , 3)/( n ^3) 3 n (b) Consider the same process as in part (a), but now we stop when the first three bins (i.e., bins 1, 2 and 3) are all occupied. The expected number of balls thrown is now

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Computer Science 174 - Fall 1998 - Sinclair - Final -...

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