Transcript Ch.15 – Probability Rules! (I Totally Agree! )

• Venn Diagram – A visual display
of a sample space; very helpful
when looking at several
variables at the same time!
Deal or No Deal!
• Suppose you are a contestant on Deal or
No Deal…
You have 20 briefcases to select from
labeled with the numbers 1 – 20 (This is
your sample space!)
• Each briefcase has an equally likely
chance of selection.
• Let’s figure out a couple of probabilities to
get started…
• P(A) =
_(count of desired outcomes)_
(count of all possible outcomes)
1
2
3
4
5
9
13
17
6
10
14
18
15
19
16
20
7
11
8
12
DEAL OR NO DEAL SAMPLE SPACE!
General Addition Rule
• For any two events A & B, the
probability of A or B is
P( A B )  P( A)  P( B )  P( A B )
Ex) Police report that 78% of drivers stopped on
suspicion of drunk driving are given a breath
test, 36% a blood test, and 22% both tests.
1) What is the probability that the suspect is given
a test?
2) What is the probability that the suspect gets
either a blood test or a breath test but NOT
both?
P(A or B but NOT both) =
3) What is the probability that the suspect gets
neither test?
P(neither test) = 1 – p(either test)
Conditional Probabilities
• Do you remember contingency
tables? Two psychologist s surveyed 478 children in
grades 4,5, and 6 in elementary schools in Michigan. They stratified
their sample, drawing roughly 1/3 from rural, 1/3 from suburban, and
1/3 from urban schools. They asked the students whether their
primary goal was to get good grades, to be popular, or to be good at
sports. One questions of interest was whether boys and girls at this
age had similar goals.
Grades
Popular
Sports
Total
Boy
117
50
60
227
Girl
130
91
30
251
Total
247
141
90
478
1) What is the probability of selecting a girl?
2) What is the probability of selecting a girl
whose goal is to be popular?
P(girl popular) =
3)The probability of selecting a student whose
goal is to excel at sports.
P(sports) =
• Conditional Probability – “the
probability of B given A”
• To find the probability of B
given A, you restrict your
sample space to just those
outcomes in A before finding
the probability of B.
What if we are given the information that
the selected student is a girl? Would that
change the probability that the selected
student’s goal is sports?
When we restrict our focus to girls, we look
only at the girls’ row of the table which
gives the conditional distribution of goals
given “girls”.
p(sports|girl) =
Formally, we write…
P( A B )
P( B A) 
P( A)
Now let’s look at the probabilities
from our last example and see
how it works using the actual
probabilities instead of looking at
just the counts.
General Multiplication Rule
• By rearranging the equation for
conditional probability, we have
for any two events A & B
P( A
B )  P( A) P( B A)
(The events do not have to be
independent.)
Independence: Events A and B are
independent whenever P(B|A) = P(B)
1)Is the probability of having good grades
as a goal independent of the sex of the
responding student?
P(grades|girl) = P(grades)
2) What about choosing success in sports?
Does P(sports|boy) = P(sports)?
Because these probabilities aren’t equal, we
can be pretty sure that choosing success in
sports as a goal is not independent of the
student’s sex. It appears that succeeding in
sports is more important to boys.