Download IB Math SL Notes CH 22GH

CH 22GH
G: SAMPLING WITH AND
WITHOUT REPLACEMENT
H: SETS AND VENN
DIAGRAMS
G: Sampling with and without replacement
 Sampling
 Sampling with replacement
 Sampling without replacement
Industrial Sampling
 Sampling is commonly used in the quality control of
industrial processes.



How it’s made
http://www.youtube.com/watch?v=b5uUV9z7hfg
http://www.youtube.com/watch?v=qwb2uZLSLhw
Example of quality control in the work force.
http://www.youtube.com/watch?v=8NPzLBSBzPI
Consider a box containing 3 red, 2 blue and 1 yellow marble. If we sample
two marbles, what is the probability we select BR?
 With replacement
 Without replacement
MORE!
 A box contains 3 red, 2 blue, 1 yellow
marble. Find the probability of getting two
different colours:
 If
replacement occurs.
 If replacement does not occur.
MORE!
 A box contains 3 red, 2 blue, 1 yellow marble. Find the
probability of getting two different colours:

If replacement occurs.

If replacement does not occur.
Even More!
 A bag contains 5 red and 3 blue marbles. Two
marbles are drawn simultaneously from the bag.
Determine the probability that at least one is red.
Sets and Venn Diagrams
 Venn Diagrams – way to represent data from a
sample space.


Rectangle – complete sample space U.
Circles – particular events
Example
 Roll a 6-sided die. What is the sample space U?
 U = {1, 2, 3, 4, 5, 6}. U is a set.
 If the event A is “a number less than 3”, then how
many outcomes are there?

A = {1, 2}
 The Venn diagram below illustrates the event A
within the universal set U.
n(U) = 6 and n(A) = 2, so
Set Notation
 Universal set or sample
space U
 Complement A’
 If U = {1, 2 , 3, 4, 5, 6}
and A = {2, 4, 6}, then
A’ =
{1, 3, 5}
Set Notation
denotes the intersection of sets A and B.


This sets contains all the elements common to both sets.
denotes the union of sets A and B.


This set contains all the elements belonging to A or B or both A
and B.
Disjoint Sets
 Disjoint sets are sets which do not have elements
in common.
Example
Let A be the set of all factors of 6, B be the set of all
positive even integers < 11, and
Answer
Let A be the set of all factors of 6, B be the set of all
positive even integers < 11, and
Another superb example
Another superb answer
Almost finished – just a few more
In a class of 30 students, 19 study Physics, 17 study
Chemistry, and 15 study both of these subjects.
Display this info on a Venn Diagram and determine
the probability that a randomly selected class member
studies:
a) Both subjects
b) At least one of the subjects
c) Physics but not Chemistry
d) Exactly one of the subjects
e) Neither subject
Answer
In a class of 30 students, 19 study Physics, 17 study Chemistry, and 15
study both of these subjects. Display this info on a Venn Diagram and
determine the probability that a randomly selected class member studies:
Use a Venn diagram to
Use a Venn diagram to