STAT 110 - Section 5
Lecture 21
Professor Hao Wang
University of South Carolina
Spring 2012
Chapter 18 – Probability Models
probability model – describes all possible outcomes
and says how to assign
probabilities to any collection of
outcomes
sample space – collection of all unique outcomes of
a random circumstance
event – a collection of outcomes
Coin Example
Suppose you are asked to roll a die with 6 faces.
What is the sample space?
Possible events are
• Roll is an even number
• Roll is an odd number
• Roll is 5 or 6
What about a roulette
Sample space ?
Example of events ?
Probability Rules
1. Any probability is a number between 0 and 1.
So if we observe an event A then we know
0  P( A)  1
Probability Rules
2. All possible outcomes together must have
probability 1.
• An outcome must occur on every trial.
• The sum of the probabilities for all possible
outcomes must be exactly 1.
Marital Status of a Random Sample of Women
• Consider the following assignment of probabilities
• Marital Status of a Random Sample of Women
Ages 25 to 29
Marital Status
Never married
Married
Widowed
Divorced
Probability
0.386
0.555
0.004
0.055
Marital Status of a Random Sample of Women
• Each of the probabilities is a number between
0 and 1.
The probabilities total to 1.
0.386 + 0.555 + 0.004 + 0.055 = 1
Any assignment of probabilities to all individual
outcomes that satisfies Rules 1 and 2 is
legitimate.
What does the probability of D need to be to make
this a probability model?
P(A)=0.3
A) 0.0
B) 0.1
C) 0.2
D) 0.3
E) 0.4
P(B)=0.2
P(C)=0.1
P(D)=?
Which of the following is not a possible probability
model:
A) P(A)=0.3 P(B)=0.4
P(C)=0.3
B) P(A)=0.3 P(B)=0.7
C) P(A)=1.0
D) P(A)=0.3 P(B)=0.6
P(C)=0.2
Incoherent
If a set of probabilities don’t satisfy rules 1 and 2 we
say they are incoherent.
This often occurs with someone’s personal
probabilities in complicated situations
Probability Rules
3. The probability that an event does not occur is 1
minus the probability that the event does occur.
This is known as the complement rule.
• Suppose that P(A) = .70
• Using this rule we can determine P(not A)
P(not A) = 1- P(A) = 1-.70 = .30
The event “not A” is known as the
complement of A which can be written as
A
c
• Suppose the probability of a horse winning a
race is 0.85. What is the probability of the
horse not winning?
• A. 0.85
• B. 0.15
• C 0.7
• D 0.2
Probability Rules
4. If two events have no outcomes in common, the
probability that one or the other occurs is the
sum of their individual probabilities.
If this is true then the events are said to be
disjoint.
Suppose events A and B are disjoint and you
know that P(A) = .40 and P(B) = .35.
What is the P(A or B)?
If P(A)=0.5 and P(B)=0.4 and A and B are disjoint,
then what is P(A or B)?
A) 0.1
B) 0.2
C) 0.4
D) 0.5
E) 0.9
• The probability a student is in honors math is
0.25, the probability a student is in honors
science is 0.3, and the probability a student is
in both is 0.2.
• What is the probability a student is in at least
one honors class?
Venn Diagrams
• The probability it will rain Wednesday AM is 30%.
The probability it will rain Wednesday PM is 30%.
The probability it will rain both Wednesday AM and
Wednesday PM is 10%.
What is the probability it will rain on Wednesday?
A) 20%
D) 50%
B) 30%
E) 60%
C) 40%
• Rule 5. Multiplication Rule: If two events, A and B,
are independent then P(A and B) = P(A)P(B)
• Independence means that the occurrence of event
A does not affect the occurrence of event B
• What is the probability that the first roll of a die is
even, and the second roll is odd?
• A – 1/36
• B – 1/12
• C – 1/4
• D – 1/2
Review for Mid Term II
• You have data on returns on common stocks for all
years since 1945. To show clearly how returns have
changed over time, your best choice of graph is a
A line graph
B bar graph
C pie chart
D histogram
• In our class poll, information was collected on
your height in inches. This is an example of a
____________ variable.
A categorical
B quantitative
9) This data set is best described as:
a) Skewed Left
b) Symmetric
c) Skewed Right
d) Bimodal
10) The mean of this data set is:
a) Approximately equal to the median
b) Greater than the median
c) Less than the median
d) Can’t tell from the picture
Questions 14-15 are based on the data set:
4 2 6 5 12 3 9
14) The median is:
a) 5.0
b) 6.0
c) 7.0
d) 8.0
e) 9.0
Questions 14-15 are based on the data set:
4 2 6 5 12 3 9
15) The third quartile (Q3) is:
a) 2.0
b) 3.0
c) 5.0
d) 7.5
e) 9.0
16) A list of 20 exam scores range from 64 to 98. If a
typo was made and the 64 was entered as a 4, then
a) The mean would become smaller and the median
would stay the same
b) The mean would stay the same and the median would
become smaller
c) Both the mean and the median would stay the same
d) Both the mean and the median would become smaller
20) For a standard normal distribution, what can we say
about the percentage of the data is within two
standard deviations of the mean?
a) 68%
b) at least 75%
c) at least 88.9%
d) 95%
e) Can’t tell from the given information
Questions 23-24 refer to the heights of a group of men
that are approximately normally distributed with a
mean of 70 inches and a standard deviation of 3 inches.
23) Approximately what percentage of men in this group
are between 73 inches and 76 inches?
a) 13.5%
b) 27%
c) 34%
d) 47.5%
e) 81.5%
Questions 23-24 refer to the heights of a group of men
that are approximately normally distributed with a
mean of 70 inches and a standard deviation of 3 inches.
24) What percentile would someone in this group who
was 67 inches tall be?
a) 0.15%
b) 2.5%
c) 16%
d) 50%
e) 84%
observatio n - mean
standard deviation
25) Physics classes are notorious for having very low
grades on their exams (that get curved at the end of
the semester). The first exam’s scores in one
semester were approximately normally distributed
with a mean of 40 and a standard deviation of 10.
The second exam’s scores were approximately
normally distributed with a mean of 30 and a standard
observatio n - mean
deviation of 5. Use the
z-scores = standard deviation to
determine which would be a better score, a 50 on
exam 1 or a 40 on exam 2?
a) 50 on exam 1
b) 40 on exam 2
c) The two scores are equivalent
• Suppose an algebra professor found that the correlation
between study time (in hours) and exam score (out of 100) is
+.80, and the regression line was found to be y = 20 + 4x. He
arrived at this equation through years of collecting data on his
students, all of whom reported studying anywhere from 0 to
20 hours for his exams.
For which values of study time does the professor’s regression
equation make sense in terms of predicting exam scores?
a. Between 0 and 20 hours.
b. Between 0 and 100 hours.
c. Anything greater than or equal to 0 hours.
d. It is not possible to predict exam score with study time.