STAT 110 - Section 5
Lecture 23
Professor Hao Wang
University of South Carolina
Spring 2012
Histogram of Mid-Term 2 Grades
min 6
Q1 17
Med:18
Q3 20
Max 24
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.
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) = .38
• Using this rule we can determine P(not A)
P(not A) = 1- P(A) = 1-.38 = .62
The event “not A” is known as the
complement of A which can be written as
A
c
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)?
Venn Diagrams
Multiplication Rule
5. Multiplication Rule: If two events 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
Probability Models for Sampling
sampling distribution – tells what values a statistic
takes in repeated samples
from the same population
and how often it takes those
values
We can’t predict the outcome of one sample, but
the outcomes of many samples have a regular
pattern.
The distribution of the statistic tells us what values
it can take and how often it takes those values
Imagine a biased coin that has a 60% chance of
coming up heads.
Flip the coin twice.
What is the distribution of the percent heads (
we see?
pˆ
)
Tree diagram
What is the
distribution of the
percent of flips that
are heads ( ) ?
What is the
distribution of the
percent of flips that
are heads ( ) ?
What is the
distribution of
the percent of
flips that are
heads ( ) ?
So the sampling distribution of
P(
pˆ )
pˆ
is:
0.0
0.5
1.0
0.16
0.48
0.36
The sampling distribution of
P(
pˆ )
pˆ is:
0.0
0.5
1.0
0.16
0.48
0.36
So, in this case, even though we’ll usually see 50% or
100% heads, we shouldn’t be surprised to see 0%
heads. That still happens 16% of the time.
But what if we flipped the coin a lot more?
Imagine a biased coin that has a 60% chance of
coming up heads.
Flip the coin 1,000 times.
What is the distribution of the percent heads (
we see?
pˆ )
If we flipped the coin 1,000 times the tree diagram
would have 1,000 columns of branches!!!!!!!!
Instead we could use the computer to simulate it.
Here are the
results of one
simulation where
the computer
repeats the
experiment 500
times.
On average the
percent
is close to 0.60,
but the spread is
pretty small (just
between 0.55
and 0.65).
And it looks
close to normal!
Example on sampling distribution
A simple random sample of 501 teens is asked,
“Do you approve or disapprove of legal gambling or
betting?”
Suppose that the true proportion that will say
“yes” is .50.
The sample proportion who say “Yes” will vary in
repeated samples according to a normal
distribution with mean 0.5 and standard deviation
of about 0.0223.
The sample proportion who say “Yes” will vary in
repeated samples according to a normal distribution
with mean 0.5 and standard deviation of about 0.022.
•What’s the probability that less than .478 say
“Yes”?
• What’s the probability that .522 or more will say
“Yes”?
• What’s the probability that 0.60 or more will say
“Yes”?
Example on sampling distribution
A simple random sample of 1000 college students is
taken and asked if they agree that “The dining choices
on my campus could be better.
If the true percent for the whole population is 60%, the
percent in the sample who will say “Yes” should be close
to a normal distribution (by the central limit theorem) with
mean 0.6 and standard deviation of about 0.0155.
Would you be surprised if 65% of the sample said “Yes”?
Would you be surprised if 65% of the sample said “Yes”?
That would be
surprising, because it
should only happen
about 6 in 10,000 times!
Example on tree diagram
There is an 80% chance you will make the first red
light driving in to work.
If you make the first, there is a 90% chance you will
also make the second.
If you miss the first, there is a 90% chance you will
also miss the second.
What is the chance you get stopped by both lights?
A) 0.08 = 8%
D) 0.18 = 18%
B) 0.09 = 9%
E) 0.72 = 72%
C) 0.16 = 16%