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11.7 Probability  (Page 4/18)

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

  5 6  

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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  
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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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