#### Transcript Chapter 5

```Section 5.3
Conditional Probability: What’s the
Probability of A, Given B?
Agresti/Franklin Statistics, 1 of 87
Conditional Probability

For events A and B, the conditional
probability of event A, given that
event B has occurred is:
P( A andB)
P (A | B) 
P( B)
Agresti/Franklin Statistics, 2 of 87
Conditional Probability
Agresti/Franklin Statistics, 3 of 87
Example: What are the Chances
of a Taxpayer being Audited?
Agresti/Franklin Statistics, 4 of 87
Example: Probabilities of a
Taxpayer Being Audited
Agresti/Franklin Statistics, 5 of 87
Example: Probabilities of a
Taxpayer Being Audited

What was the probability of being
audited, given that the income was ≥
\$100,000?
• Event A:
• Event B:
Taxpayer is audited
Taxpayer’s income ≥ \$100,000
Agresti/Franklin Statistics, 6 of 87
Example: Probabilities of a
Taxpayer Being Audited
P(A and B) 0.0010
P(A| B) 

 0.007
P(B)
0.1334
Agresti/Franklin Statistics, 7 of 87
Example: The Triple Blood Test
for Down Syndrome

A positive test result states that the
condition is present

A negative test result states that the
condition is not present
Agresti/Franklin Statistics, 8 of 87
Example: The Triple Blood Test
for Down Syndrome

False Positive: Test states the
condition is present, but it is actually
absent

False Negative: Test states the
condition is absent, but it is actually
present
Agresti/Franklin Statistics, 9 of 87
Example: The Triple Blood Test
for Down Syndrome

A study of 5282 women aged 35 or
over analyzed the Triple Blood Test to
test its accuracy
Agresti/Franklin Statistics, 10 of 87
Example: The Triple Blood Test
for Down Syndrome
Agresti/Franklin Statistics, 11 of 87
Example: The Triple Blood Test
for Down Syndrome

Assuming the sample is representative
of the population, find the estimated
probability of a positive test for a
randomly chosen pregnant woman 35
years or older
Agresti/Franklin Statistics, 12 of 87
Example: The Triple Blood Test
for Down Syndrome

P(POS) = 1355/5282 = 0.257
Agresti/Franklin Statistics, 13 of 87
Example: The Triple Blood Test
for Down Syndrome

Given that the diagnostic test result is
positive, find the estimated
probability that Down syndrome truly
is present
Agresti/Franklin Statistics, 14 of 87
Example: The Triple Blood Test
for Down Syndrome
P(D and POS) 48 / 5282
P(D| POS) 


P(POS)
1355/ 5282
0.009
 0.035
0.257
Agresti/Franklin Statistics, 15 of 87
Example: The Triple Blood Test
for Down Syndrome

Summary: Of the women who tested
positive, fewer than 4% actually had
fetuses with Down syndrome
Agresti/Franklin Statistics, 16 of 87
Multiplication Rule for Finding
P(A and B)

For events A and B, the probability
that A and B both occur equals:
• P(A and B) = P(A|B) x P(B)
•
also
P(A and B) = P(B|A) x P(A)
Agresti/Franklin Statistics, 17 of 87
Example: How Likely is a Double
Fault in Tennis?

Roger Federer – 2004 men’s
champion in the Wimbledon tennis
tournament
• He made 64% of his first serves
• He faulted on the first serve 36% of the
•
time
Given that he made a fault with his first
serve, he made a fault on his second serve
only 6% of the time
Agresti/Franklin Statistics, 18 of 87
Example: How Likely is a Double
Fault in Tennis?

Assuming these are typical of his
serving performance, when he serves,
what is the probability that he makes
a double fault?
Agresti/Franklin Statistics, 19 of 87
Example: How Likely is a Double
Fault in Tennis?



P(F1) = 0.36
P(F2|F1) = 0.06
P(F1 and F2) = P(F2|F1) x P(F1)
= 0.06 x 0.36 = 0.02
Agresti/Franklin Statistics, 20 of 87
Sampling Without
Replacement

Once subjects are selected from a
population, they are not eligible to be
selected again
Agresti/Franklin Statistics, 21 of 87
Example: How Likely Are You to
Win the Lotto?

In Georgia’s Lotto, 6 numbers are
randomly sampled without
replacement from the integers 1 to 49

You buy a Lotto ticket. What is the
probability that it is the winning
ticket?
Agresti/Franklin Statistics, 22 of 87
Example: How Likely Are You to
Win the Lotto?

P(have all 6 numbers) = P(have 1st and 2nd
and 3rd and 4th and 5th and 6th)
= P(have 1st)xP(have 2nd|have 1st)xP(have 3rd|
have 1st and 2nd) …P(have 6th|have 1st, 2nd,
3rd, 4th, 5th)
Agresti/Franklin Statistics, 23 of 87
Example: How Likely Are You to
Win the Lotto?
6/49 x 5/48 x 4/47 x 3/46 x 2/45 x 1/44
= 0.00000007
Agresti/Franklin Statistics, 24 of 87
Independent Events Defined
Using Conditional Probabilities

Two events A and B are independent
if the probability that one occurs is
not affected by whether or not the
other event occurs
Agresti/Franklin Statistics, 25 of 87
Independent Events Defined
Using Conditional Probabilities

Events A and B are independent if:
P(A|B) = P(A)

If this holds, then also P(B|A) = P(B)

Also, P(A and B) = P(A) x P(B)
Agresti/Franklin Statistics, 26 of 87
Checking for Independence

Here are three ways to check whether
events A and B are independent:
• Is P(A|B) = P(A)?
• Is P(B|A) = P(B)?
• Is P(A and B) = P(A) x P(B)?

If any of these is true, the others are also
true and the events A and B are
independent
Agresti/Franklin Statistics, 27 of 87
Example: How to Check Whether
Two Events are Independent

The diagnostic blood test for Down
syndrome:
POS = positive result
NEG = negative result
D = Down Syndrome
DC = Unaffected
Agresti/Franklin Statistics, 28 of 87
Example: How to Check Whether
Two Events are Independent
Blood Test:
Status
POS
NEG
Total
D
0.009
0.001
0.010
Dc
0.247
0.742
0.990
Total
0.257
0.743
1.000
Agresti/Franklin Statistics, 29 of 87
Example: How to Check Whether
Two Events are Independent

Are the events POS and D
independent or dependent?
• Is P(POS|D) = P(POS)?
Agresti/Franklin Statistics, 30 of 87
Example: How to Check Whether
Two Events are Independent

Is P(POS|D) = P(POS)?

P(POS|D) =P(POS and D)/P(D)
= 0.009/0.010 = 0.90

P(POS) = 0.256

The events POS and D are
dependent
Agresti/Franklin Statistics, 31 of 87
Section 5.4
Applying the Probability Rules
Agresti/Franklin Statistics, 32 of 87
Is a “Coincidence” Truly an
Unusual Event?

The law of very large numbers states
that if something has a very large
number of opportunities to happen,
occasionally it will happen, even if it
seems highly unusual
Agresti/Franklin Statistics, 33 of 87
Example: Is a Matching Birthday
Surprising?

What is the probability that at least
two students in a group of 25
students have the same birthday?
Agresti/Franklin Statistics, 34 of 87
Example: Is a Matching Birthday
Surprising?

P(at least one match) = 1 – P(no
matches)
Agresti/Franklin Statistics, 35 of 87
Example: Is a Matching Birthday
Surprising?

P(no matches) = P(students 1 and 2
and 3 …and 25 have different
birthdays)
Agresti/Franklin Statistics, 36 of 87
Example: Is a Matching Birthday
Surprising?

P(no matches) =
(365/365) x (364/365) x (363/365) x …
x (341/365)

P(no matches) = 0.43
Agresti/Franklin Statistics, 37 of 87
Example: Is a Matching Birthday
Surprising?

P(at least one match) =
1 – P(no matches) = 1 – 0.43 = 0.57
Agresti/Franklin Statistics, 38 of 87
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