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### Selecting a Student Council

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Date: 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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Associated Topics:
High School Probability

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