Download The Addition Rule - MATH 130, Elements of Statistics I

The Addition Rule
MATH 130, Elements of Statistics I
J. Robert Buchanan
Department of Mathematics
Fall 2015
Venn Diagrams
A Venn diagram represents the sample space of an
experiment as a rectangle and events as circles.
S
C
A
B
Example (1 of 2)
The demographics of a sample of students are described in the
table below. Draw a Venn diagram representing the relationship
between a student being female and the student being enrolled
in the school of education.
Gender
Male
Female
School Division
Education
Science/Math
Humanities
Education
Science/Math
Humanities
Frequency
55
65
50
75
30
50
Example (2 of 2)
Students
115
80
Female
55
75
Education
Each region of the Venn diagram contains the number of
students belonging to that demographic category.
Mutually Exclusive Events
Definition
Two events are mutually exclusive (or disjoint) if they have no
outcomes in common.
Mutually Exclusive Events
Definition
Two events are mutually exclusive (or disjoint) if they have no
outcomes in common.
Theorem (Addition Rule for Disjoint Events)
If A and B are disjoint events then
P(A or B) = P(A) + P(B).
Example (1 of 4)
Using the demographic data given in the table below find the
probabilities asked for in the following slides.
Gender
Male
Female
School Division
Education
Science/Math
Humanities
Education
Science/Math
Humanities
Frequency
55
65
50
75
30
50
Example (2 of 4)
Find the probability that a randomly selected student is a male
education major or a male science major.
Example (2 of 4)
Find the probability that a randomly selected student is a male
education major or a male science major.
Students
155
55
Education
65
ScienceMath
Humanities
50
P(M. Ed. or M. Sci.) = P(M. Ed.)+P(M. Sci.) =
55
65
+
= 0.369
325 325
Example (3 of 4)
Find the probability that a randomly selected student is a male
education major or a female science major.
Example (3 of 4)
Find the probability that a randomly selected student is a male
education major or a female science major.
Students
240
Female Science
30
Male Education
55
P(M. Ed. or F. Sci.) = P(M. Ed.)+P(F. Sci.) =
55
30
+
= 0.262
325 325
Example (4 of 4)
Find the probability that a randomly selected student is a
humanities major or an education major.
Example (4 of 4)
Find the probability that a randomly selected student is a
humanities major or an education major.
Students
Humanities
100
Education
130
ScienceMath
95
P(Hum. or Ed.) = P(Hum.) + P(Ed.) =
100 130
+
= 0.708
325 325
Benford’s Law
The symbols in our number system are
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. After studying a great deal of data
physicist Frank Benford discovered that some symbols occur
more frequently as the first digit of a number.
Benford’s Law
The symbols in our number system are
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. After studying a great deal of data
physicist Frank Benford discovered that some symbols occur
more frequently as the first digit of a number.
Digit
Rel. Freq.
1
0.301
2
0.176
3
0.125
4
0.097
5
0.079
6
0.067
7
0.058
8
0.051
9
0.046
Benford’s Law
The symbols in our number system are
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. After studying a great deal of data
physicist Frank Benford discovered that some symbols occur
more frequently as the first digit of a number.
Digit
Rel. Freq.
1
0.301
2
0.176
3
0.125
4
0.097
5
0.079
6
0.067
7
0.058
8
0.051
9
0.046
1. Verify that Benford’s Law is a probability model.
2. Determine the probability that a randomly selected first
digit of a number is 5 or 6.
3. Determine the probability that a randomly sleected first
digit of a number is less than 5.
Solution
1. Probability model:
0.301 + 0.176 + 0.125 + 0.097 + 0.079 + 0.067 + 0.058 + 0.051 + 0.046 = 1
2. P(5 or 6) =
3. P(less than 5) =
Solution
1. Probability model:
0.301 + 0.176 + 0.125 + 0.097 + 0.079 + 0.067 + 0.058 + 0.051 + 0.046 = 1
2. P(5 or 6) = P(5) + P(6) = 0.079 + 0.067 = 0.146
3. P(less than 5) =
Solution
1. Probability model:
0.301 + 0.176 + 0.125 + 0.097 + 0.079 + 0.067 + 0.058 + 0.051 + 0.046 = 1
2. P(5 or 6) = P(5) + P(6) = 0.079 + 0.067 = 0.146
3. P(less than 5) = P(1) + P(2) + P(3) + P(4) = 0.699
General Addition Rule
Theorem (General Addition Rule)
For any two events A and B,
P(A or B) = P(A) + P(B) − P(A and B).
Example
Consider the contingency table below describing a sample of
college students.
Female
Male
Total
Full time
80
60
140
Part time
40
20
60
Total
120
80
200
Find the probability that a randomly selected student is
1. a student is part time or female,
2. full time or female,
3. male.
Complements
Definition
Let S denote the sample space of an experiment and let E
denote an event. The complement of E, denoted E c , is all the
outcomes in the sample space that are not outcomes in event
E.
Theorem (Complement Rule)
If E represents any event and E c represents the complement of
E, then
P(E c ) = 1 − P(E).
Example (1 of 3)
A pair of fair dice are rolled. What is the probability that the sum
shown on the dice is at least 3?
Example (1 of 3)
A pair of fair dice are rolled. What is the probability that the sum
shown on the dice is at least 3?
P(E ≥ 3) = 1 − P(E = 2) = 1 −
1
35
=
= 0.972
36
36
Example (2 of 3)
Out of 150,000 high school seniors who played basketball,
3800 made their college team. Only 2400 played as seniors in
college. Sixty-four played professional basketball. What is the
probability that a high school basketball player does not play as
a college senior? (A Venn diagram may be helpful.)
Example (3 of 3)
Players
146,200
College Sr.
Pros
2336
64
College
1400
p =1−
2400
= 0.984
150000
Drawing Cards
Consider randomly selecting a card from a standard 52-card
deck. What is the probability of:
1. drawing a queen?
2. drawing a diamond or a queen?
3. drawing a seven or a queen?