<< Chapter < Page Chapter >> Page >

Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than 10.

5 6

Got questions? Get instant answers now!

Computing probability using counting theory

Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.

Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are C ( 5 , 2 ) ways to select 2 phones that are not defective. There are 8 phones, so there are C ( 8 , 2 ) ways to select 2 phones. The probability of selecting 2 phones that are not defective is:

ways to select 2 phones that are not defective ways to select 2 phones = C ( 5 , 2 ) C ( 8 , 2 )                                                                          = 10 28                                                                          = 5 14

Computing probability using counting theory

A child randomly selects 5 toys from a bin containing 3 bunnies, 5 dogs, and 6 bears.

  1. Find the probability that only bears are chosen.
  2. Find the probability that 2 bears and 3 dogs are chosen.
  3. Find the probability that at least 2 dogs are chosen.
  1. We need to count the number of ways to choose only bears and the total number of possible ways to select 5 toys. There are 6 bears, so there are C ( 6 , 5 ) ways to choose 5 bears. There are 14 toys, so there are C ( 14 , 5 ) ways to choose any 5 toys.
    C ( 6 , 5 ) C ( 14 , 5 ) = 6 2 , 002 = 3 1 , 001
  2. We need to count the number of ways to choose 2 bears and 3 dogs and the total number of possible ways to select 5 toys. There are 6 bears, so there are C ( 6 , 2 ) ways to choose 2 bears. There are 5 dogs, so there are C ( 5 , 3 ) ways to choose 3 dogs. Since we are choosing both bears and dogs at the same time, we will use the Multiplication Principle. There are C ( 6 , 2 ) C ( 5 , 3 ) ways to choose 2 bears and 3 dogs. We can use this result to find the probability.
    C ( 6 , 2 ) C ( 5 , 3 ) C ( 14 , 5 ) = 15 10 2 , 002 = 75 1 , 001
  3. It is often easiest to solve “at least” problems using the Complement Rule. We will begin by finding the probability that fewer than 2 dogs are chosen. If less than 2 dogs are chosen, then either no dogs could be chosen, or 1 dog could be chosen.

    When no dogs are chosen, all 5 toys come from the 9 toys that are not dogs. There are C ( 9 , 5 ) ways to choose toys from the 9 toys that are not dogs. Since there are 14 toys, there are C ( 14 , 5 ) ways to choose the 5 toys from all of the toys.

    C ( 9 , 5 ) C ( 14 , 5 ) = 63 1 , 001

    If there is 1 dog chosen, then 4 toys must come from the 9 toys that are not dogs, and 1 must come from the 5 dogs. Since we are choosing both dogs and other toys at the same time, we will use the Multiplication Principle. There are C ( 5 , 1 ) C ( 9 , 4 ) ways to choose 1 dog and 1 other toy.

    C ( 5 , 1 ) C ( 9 , 4 ) C ( 14 , 5 ) = 5 126 2 , 002 = 315 1 , 001

    Because these events would not occur together and are therefore mutually exclusive, we add the probabilities to find the probability that fewer than 2 dogs are chosen.

    63 1 , 001 + 315 1 , 001 = 378 1 , 001

    We then subtract that probability from 1 to find the probability that at least 2 dogs are chosen.

    1 378 1 , 001 = 623 1 , 001
Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 9

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask