1. Home
  2. Algebra and trigonometry
  3. Sequences, probability, and
  4. Probability

13.7 Probability  (Page 4/18)

<< Chapter < Page
  Algebra and trigonometry     Page 4 / 18
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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 9

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask