{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

hw1sol

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

This preview shows pages 1–2. Sign up to view the full content.

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).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern