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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

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
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bill
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bill
-24m+3+3mÁ^2
Susan
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Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
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Aphelele
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Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
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Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
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Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
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Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
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Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
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Method
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Amoon
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Enock
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Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
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When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
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Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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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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