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assignment2sol - ELEN 303 Assignment 2 Instructor Email Oce...

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ELEN 303: Assignment 2 Instructor: Dr. Jean-Fran¸ cois Chamberland Email: [email protected] (Subject ECEN 303) Office: Room 244F WERC Office Hours: Tue 2:30 - 3:45 p.m. Problems: 1. Suppose that there are n students in a room, all born in 1985 (a non-leap year). What is the probability that no two of them celebrate their birthday on the same day of the year? How large does n need to be so that this probability becomes less than 0.5? Every student in the room celebrates his or her birthday on one of the 365 days in a year. Therefore there is a total of (365) n possible outcomes. Assuming that each of these outcome is equally likely, we can compute the desired probability as 365 · 364 · · · (365 n + 1) 365 n = 365! (365 n )!365 n . Furthermore, when n 23, the probability that no two students celebrate their birthday on the same day is less than 0.5. 2. (FCP: 2.1) A box contains 3 marbles, 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box, then replacing it in the box and drawing a second marble from the box. Describe the sample space. Repeat when the second marble is drawn without first replacing the first marble. In the first case, the sample space is a finite set that contains 9 outcomes. These outcomes can be represented as Ω = { ( r,r ) , ( r,g ) , ( r,b ) , ( g,r ) , ( g,g ) , ( g,b ) , ( b,r ) , ( b,g ) , ( b,b ) } ; the pair ( i,j ) represents a drawing where the color of the first marble is i , and the color of the second marble is j . In the second experiment described above, the sample space has 6 outcomes. Using the
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