<< Chapter < Page Chapter >> Page >

You are president of an on-campus special events organization. You need a committee of seven students to plan a special birthday party for the president of the college. Your organization consists of 18 women and 15 men. You are interested in the number of men on your committee. If the members of the committee are randomly selected, what is the probability that your committee has more than four men?

This is a hypergeometric problem because you are choosing your committee from two groups (men and women).

a. Are you choosing with or without replacement?

a. without

Got questions? Get instant answers now!

b. What is the group of interest?

b. the men

Got questions? Get instant answers now!

c. How many are in the group of interest?

c. 15 men

Got questions? Get instant answers now!

d. How many are in the other group?

d. 18 women

Got questions? Get instant answers now!

e. Let X = _________ on the committee. What values does X take on?

e. Let X = the number of men on the committee. x = 0, 1, 2, …, 7.

Got questions? Get instant answers now!

f. The probability question is P (_______).

f. P ( x >4)

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Try it

A palette has 200 milk cartons. Of the 200 cartons, it is known that ten of them have leaked and cannot be sold. A stock clerk randomly chooses 18 for inspection. He wants to know the probability that among the 18, no more than two are leaking. Give five reasons why this is a hypergeometric problem.

  1. There are two groups.
  2. You are concerned with a group of interest.
  3. You sample without replacement.
  4. Each pick is not independent.
  5. You are not dealing with Bernoulli Trials.
Got questions? Get instant answers now!

Notation for the hypergeometric: h = hypergeometric probability distribution function

X ~ H ( r , b , n )

Read this as " X is a random variable with a hypergeometric distribution." The parameters are r , b , and n ; r = the size of the group of interest (first group), b = the size of the second group, n = the size of the chosen sample.

A school site committee is to be chosen randomly from six men and five women. If the committee consists of four members chosen randomly, what is the probability that two of them are men? How many men do you expect to be on the committee?

Let X = the number of men on the committee of four. The men are the group of interest (first group).

X takes on the values 0, 1, 2, 3, 4, where r = 6 , b = 5 , and n = 4 . X ~ H (6, 5, 4)

Find P ( x = 2). P ( x = 2) = 0.4545 (calculator or computer)

Note

Currently, the TI-83+ and TI-84 do not have hypergeometric probability functions. There are a number of computer packages, including Microsoft Excel, that do.

The probability that there are two men on the committee is about 0.45.

The graph of X ~ H (6, 5, 4) is:

This graph shows a hypergeometric probability distribution. It has five bars that are slightly normally distributed. The x-axis shows values from 0 to 4 in increments of 1, representing the number of men on the four-person committee. The y-axis ranges from 0 to 0.5 in increments of 0.1.

The y -axis contains the probability of X , where X = the number of men on the committee.

You would expect m = 2.18 (about two) men on the committee.

The formula for the mean is μ = n r r + b = ( 4 ) ( 6 ) 6 + 5 = 2.18

Got questions? Get instant answers now!

Try it

An intramural basketball team is to be chosen randomly from 15 boys and 12 girls. The team has ten slots. You want to know the probability that eight of the players will be boys. What is the group of interest and the sample?

The group of interest is the 15 boys. The sample consists of the ten slots on the intramural basketball team.

Got questions? Get instant answers now!

Chapter review

A hypergeometric experiment is a statistical experiment with the following properties:

  1. You take samples from two groups.
  2. You are concerned with a group of interest, called the first group.
  3. You sample without replacement from the combined groups.
  4. Each pick is not independent, since sampling is without replacement.
  5. You are not dealing with Bernoulli Trials.

The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest. The distribution of X is denoted X ~ H ( r , b , n ), where r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample. It follows that
n r + b . The mean of X is μ = n r r  +  b and the standard deviation is σ = r b n ( r  +  b  −  n ) ( r  +  b ) 2 ( r  +  b 1) .

Formula review

X ~ H ( r , b , n ) means that the discrete random variable X has a hypergeometric probability distribution with r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample.

X = the number of items from the group of interest that are in the chosen sample, and X may take on the values x = 0, 1, ..., up to the size of the group of interest. (The minimum value for X may be larger than zero in some instances.)

n r + b

The mean of X is given by the formula μ = n r r  +  b and the standard deviation is = r b n ( r  +  b n ) ( r  +  b ) 2 ( r  +  b 1) .

Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample.

In words, define the random variable X .

X = the number of business majors in the sample.

Got questions? Get instant answers now!

What values does X take on?

2, 3, 4, 5, 6, 7, 8, 9

Got questions? Get instant answers now!

Find the standard deviation.

Got questions? Get instant answers now!

On average ( μ ), how many would you expect to be business majors?

6.26

Got questions? Get instant answers now!

Questions & Answers

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introductory statistics' conversation and receive update notifications?

Ask