Transcript 3.2-3.4 PowerPoint

Assignment 3.1 due Tuesday March 4, 2014
Sta220 - Statistics
Mr. Smith
Room 310
Class #8
Section 3.2-3.4
3.2 Unions and Intersections
• The union of two events A and B is the event that
occurs if either A or B (or both) occurs on a single
performance of the experiment. We denote the
union of events A and B by the symbol A U B.
• The intersection of two events A and B is the
event that occurs if both A and B occur on a
single performance of the experiment. We write
𝐴 ∩ 𝐵 for the intersection of A and B.
Venn diagrams for union and intersection
Copyright © 2013 Pearson
Education, Inc.. All rights
reserved.
Problem:
Consider a die-toss experiment in which the
following events are defined:
A: {Toss an even number.}
B: {Toss a number less than or equal to 3.}
a) Describe 𝐴 ∪ 𝐵 for this experiment.
b) Describe 𝐴 ∩ 𝐵 for this experiment.
c) Calculate 𝑃(𝐴 ∪ 𝐵) and P(𝐴 ∩ 𝐵),assuming that
the die is fair.
Figure 3.8 Venn diagrams for die toss
Copyright © 2013 Pearson
Education, Inc.. All rights
reserved.
Solution
a. 𝐴 ∪ 𝐵 = {1, 2, 3, 4, 6}
b. 𝐴 ∩ 𝐵 = 2
c. 𝑃 𝐴 ∪ 𝐵 = 𝑃 1 + 𝑃 2 + 𝑃 3 + 𝑃 4 + 𝑃 6
=
𝑃 𝐴∩𝐵 =
1
6
1
1
1
1
+ + + +
6
6
6
6
1
𝑃 2 =
6
=
5
6
Note: Unions and intersections can be defined
for more than two events. For example, the
event A U B U C represents the union of three
events: A, B, and C. This event, which includes
the set of sample points in A, B, or C, will occur
if any one (or more) of the events A, B, and C
occurs. Similarly, the intersection 𝐴 ∩ 𝐵 ∩ 𝐶 is
the event that all three of the events A, B, and C
occur.
Problem:
Family Planning Perspectives reported on a
study of over 200,000 births in New Jersey over
a recent two-year period. The study investigated
the link between the mother’s race and the age
at which she gave birth (called maternal age).
The percentages of the total number of births in
New Jersey, by the maternal age and race
classifications, are given in Table 3.4.
Table 3.4
Percentage of New Jersey Birth Mothers, by Age and Race
Race
Maternal Age (Years) White
White
Black
≤ 𝟏𝟕
2%
2%
18-19
3%
2%
20-29
41%
12%
≥ 𝟑𝟎
33%
5%
This table is called a two-way table, since responses are
classified according to two variables: maternal age (rows)
and race (columns).
Define the following event:
A: {A New Jersey birth mother is white.}
B: {A New Jersey mother was a teenager when
giving birth.}
a) Find P(A) and P(B).
b) Find P(𝐴 ∪ 𝐵) .
c) Find 𝑃(𝐴 ∩ 𝐵).
First Identify you sample points
𝐸1 : {≤ 17𝑦𝑟𝑠, 𝑤ℎ𝑖𝑡𝑒}
𝐸2 : {18 − 19𝑦𝑟𝑠, 𝑤ℎ𝑖𝑡𝑒}
𝐸3 : {20 − 29𝑦𝑟𝑠, 𝑤ℎ𝑖𝑡𝑒}
𝐸4 : {≥ 30 𝑦𝑟𝑠, 𝑤ℎ𝑖𝑡𝑒
𝐸5 : {≤ 17𝑦𝑟𝑠, 𝑏𝑙𝑎𝑐𝑘}
𝐸6 : {18 − 19 𝑦𝑟𝑠, 𝑏𝑙𝑎𝑐𝑘}
𝐸7 : {20 − 29𝑦𝑟𝑠, 𝑏𝑙𝑎𝑐𝑘}
𝐸8 : {≥ 30𝑦𝑟𝑠, 𝑏𝑙𝑎𝑐𝑘}
𝑃(𝐸1 ) = 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑏𝑖𝑟𝑡ℎ 𝑚𝑜𝑡ℎ𝑒𝑟𝑠 𝑖𝑛 𝑎𝑔𝑒 −
𝑟𝑎𝑐𝑒 𝑐𝑙𝑎𝑠𝑠 {≤ 17𝑦𝑟𝑠, 𝑤ℎ𝑖𝑡𝑒}= .02
𝑃(𝐸2 ) =.03
𝑃(𝐸3 ) =.41
𝑃(𝐸4 ) =.33
𝑃(𝐸5 ) =.02
𝑃(𝐸6 ) =.02
𝑃(𝐸7 ) =.12
𝑃(𝐸8 ) =.05
Solution
On the White Board
3.3 Complementary Events
The complement of an event A is the event that
A does not occur -that is, the event consisting of
all sample points that are not in event A. We
denote the complement of A by 𝐴𝑐
RULE of COMPLEMENTS
An event A is a collection of sample points, and
the sample points included in 𝐴𝑐 are those not in
A.
𝑃(𝐴) + 𝑃(𝐴𝑐 ) = 1
Problem:
Consider the experiment of tossing fair coins.
Define the following event:
A: {Observing at least one head}.
a) Find P(A) if 2 coins are tossed.
b) Find P(A) if 10 coins are tossed.
Solution
On the White Board
3.4 The Additive Rule and Mutually
Exclusive Events
Additive Rule of Probability
The probability of the union of events A and B is
the sum of the probability of event A and the
probability of event B, minus the probability of
the intersection of events A and B; that is
𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)
Problem
Hospital records show that 12% of all patients
are admitted for surgical treatment, 16% are
admitted for obstetrics, and 2% receive both
obstetrics and surgical treatment. If a new
patient is admitted to the hospital, what is the
probability that the patient will be admitted for
surgery, for obstetrics, or for both?
Solution
Consider the following events:
A: {A patient admitted to the hospital receives
surgical treatment}
B:{A patient admitted to the hospital receives
obstetrics treatment}
Solution
P(A) = .12
P(B) = .16
𝑃 𝐴 ∩ 𝐵 = .02
What is 𝑃 𝐴 ∪ 𝐵 = ?
Events A and B are mutually exclusive if 𝐴 ∩ 𝐵
contains no sample points – that is, if A and B
have no sample points in common. For mutually
exclusive events,
𝑃 𝐴∩𝐵 =0
Probability of Union of Two Mutually
Exclusive Events
If two events A and B are mutually exclusive, the
probability of the union of A and B equals the
sum of the probability of A and the probability
of B; that is 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃(𝐵).
CAUTION: The preceding formula is false if the events are not mutually
exclusive. In that case (i.e. two non-mutually exclusive events), you must
apply the general additive rule of probability.
Problem
Consider the experiment of tossing two
balanced coins. Find the probability of observing
at least one head.
Solution
Define the events
A: {Observe at least one head}
B: {Observe exactly one head}
C: {Observe exactly two heads}
Note A = 𝐵 ∪ 𝐶
Assignment 3.2-3.4 due Friday March 7, 2014 at
the beginning of class. This assignment is worth
45 points.
Make sure you read the directions.