Selecting a Student CouncilDate: 10/08/97 at 22:29:58 From: Bhavika Subject: Probability Dear Dr. Math, A school student council is to be formed consisting of 10 students selected from Year 11 and Year 12 classes. There are 120 students for year 11 and 80 year-12 students. A) What is the probability of a student being selected in each of the following cases: Q1. Each student has the same chance of being selected. Q2. The same number of Year 11 And Year 12 students are to be selected. Q3. The numbers selected in Yr. 11 and Yr. 12 are to be in proportion to the numbers in the classes. B) How would you select the students at random ? C) From the student council there is to be a president, 2 vice presidents, and a secretary. Discuss the probability of any student being chosen for those positions. Eagerly awaiting the answers.... Keiko Goto Date: 10/09/97 at 08:15:07 From: Doctor Anthony Subject: Re: Probability > Q1. Each student has the same chance of being selected. A quick answer is to say that any person has 10 chances in 200, giving a probability 10/200 = .05 It is instructive, however, to look at a more general method for this type of question. The probability that student A is selected is calculated from C(1,1) x C(199,9) ----------------- = 0.05 C(200,10) Here C(199,9), for example, means the number of ways that 9 people could be chosen from 199. So to get student A we must choose 1 from 1, and 9 from 199. The number of possible groups of 10 is C(200,10) and this is the denominator of the probability. > Q2. The same number of Year 11 And Year 12 students are to be selected. In year 11 your chance of being selected is: C(1,1) x C(119,4) x C(80,5) --------------------------- = 0.0085 C(200,10) In year 12 your chance of being selected is: C(1,1) x C(120,5) x C(79,4) --------------------------- = 0.01275 C(200,10) > Q3. The numbers selected in Yr. 11 and Yr. 12 are to be in proportion to the numbers in the classes. We have a 3:2 ratio of 11th year to 12 th year students, i.e 3/5 of the 10 will be year 11, 2/5 will be year 12. That is 6 from year 11 and 4 from year 12. In year 11 your chance of being selected is C(1,1) x C(119,5) x C(80,4) --------------------------- = 0.012866 C(200,10) In year 12 your chance of being selected is C(1,1) x C(120,6) x C(79,3) --------------------------- = 0.012866 C(200,10) >B) How would you select the students at random ? Give each student a number from 1 to 200. Use a random number generator to choose 10 numbers. Alternatively, pull 10 numbers from a hat after giving them a very thorough stir. >C) From the student council there is to be a president, 2 vice >presidents, and a secretary. Discuss the probability of any student >being chosen for those positions. In practice these positions would not be random. However if they decide to choose 4 people at random, and the order in which the names appear will be the order in which the posts of president, 2 vice- presidents and secretary are allocated, the probabilities of being selected are as follows. (1) president = 1/10 (2) vice president = Your name could come up in second or third place. = no, yes + no, no, yes = 9/10 x 1/9 + 9/10 x 8/9 x 1/8 = 1/10 + 1/10 = 1/5 (3) secretary = Your name must come up in 4th position = no, no, no, yes = 9/10 x 8/9 x 7/8 x 1/7 = 1/10 -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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