Question 1072879: A container contains 10 diesel engines. The company chooses 6 engines at random, and will not ship the container if any of the engines chosen are defective. Find the probability that a container will be shipped even though it contains 2 defectives if the sample size is 6.
Can someone please breakdown the steps to solve this. This is an online class and the help feature isn't explaining how to get the numbers needed to solve it. Thank you
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! I am baffled by the phrase "if the sample size is 6" at the end of the problem.
The problem said before that the company chooses 6 engines out of the 10 in the container. What is that phrase about sample size for?
Anyway, there are 
different possible sets of 6 engines that can be chosen.
NOTE: That is combinations of 10 taking 6 at a time.
Your online class may use a symbol with a  for combinations,
instead of the  notation I used.
There are only 
sets of 6 good engines that can be made with the
 good engines in that container that has  defective ones.
As a fraction that is  ,
and that is the probability that the set of 6 engines chosen does not have a defective one.
FORMULAS:
I do not like formulas as a substitute for understanding concepts.
The concept that is relevant to this problem is the concept of combinations.
The number ways of choosing a set of  engines
out of a set of  engines, can be seen as
, the number of ways to pick the chosen ,
or  , the number of ways of selecting the ( not chosen.
 ,
with factors in numerator, and factors in denominator,
makes for an easier calculation than
 .
Below are the formulas and why they are the way they are.
Permutations of  items taking all is
the number of different ordered lists of items that can be made
from all items of a set of different items is
   ,
because there are
ways to pick the first item for the list,
 ways to pick the second item for the list,
 ways to pick the third item for the list,
and so on, until you have
 ways to pick one of the items not picked before,
ways to pick one of the items not picked before,
and  choice for the last item on the list.
That  is the number of options a teacher would have
to randomly line up all students in a class.
Permutations of  items taking of them is
the number of different ordered lists of items that can be made
from the different items in a set.
That number is
That is a product of factors.
That is the number of options a teacher would have to randomly select
a line of of the students in the school.
The teacher would have
choices for the first student in the line,
 choices for the second student in the line,
 choices for the third student in the line,
and so on, until there are  choices possible
for the last of the picks.
For the last of the picks,
there are students wishing (or dreading) to be the last pick.
(After that last pick, there are  students left out,
disappointed, or relieved that they were not picked).
You can see how factors in the factorials ratio simplify leaving only factors.
 .
Combinations of items taking of them is
the number of different sets of items that can be made
from the different items in a set.
The formula for combinations of taken at a time is
 .
It is the number of possible sets of the students in the school that can be made.
There are  possible lists,
but they are not going to be lined up, order does not matter.
Any set of students will be listed in a different order in lists,
but it is just set,
so the possible lists boil doen to
 possible sets.
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