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hw1sol - Math 361 X1 Homework 1 Solutions Spring 2003...

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Math 361 X1 Homework 1 Solutions Spring 2003 Graded problems: 1; 2(b);3;5; each worth 3 pts., maximal score is 12 pts. Problem 1. A coin is tossed repeatedly. What is the probability that the second head appears at the 5th toss? (Hint: Since only the first five tosses matter, you can assume that the coin is tossed only 5 times.) Solution. Since only the first five tosses matter, we can take Ω = { HHHHH, HHHHT, . . . } , i.e., the set of all 2 5 = 32 five letter sequences of H and T, with equal probabilities. The event in question corresponds to the subset A = { HTTTH, THTTH, TTHTH, TTTH } . Since #(Ω) = 32 and #( A ) = 4, this event has probability P ( A ) = #( A ) / #(Ω) = 4 / 2 5 = 1 / 8. Problem 2. [1.1:8(a),(c)] Suppose two n -sided dice are rolled. Define an appropriate probability space Ω and find the probabilities of the following events. (a) the maximum of the two numbers rolled is less than or equal to 2; (b) the maximum of the two numbers rolled is exactly equal to 3. (The maximum is the larger of the two numbers; e.g., max(3 , 5) = 5, or max(3 , 3) = 3.) Solution. Letting a 1 and a 2 denote the two numbers appearing, the possible outcomes are tuples ( a 1 , a 2 ) with each a i ranging from 1 to n . Thus, an appropriate outcome space is given by Ω = { ( a 1 , a 2 ) : a i = 1 , 2 , . . . , n } , with each outcome being equally likely. (a) The event (a) corresponds to the subset A = { ( a 1 , a 2 ) : a i = 1 , 2 } and its probability is given by P ( A ) = #( A ) #(Ω) = 2 × 2 n × n = 4 n 2 (assuming n 2).
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