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Section 5.4 - Conditional Probability
Objectives:
1.
Understand the meaning of conditional probability.
2.
Learn the general Multiplication Rule:
P(A  B)  P(A and B)  P(A)  P(B | A)
P(A  B)  P(A and B)  P(B)  P(A | B)
Section 5.4 - Conditional Probability
Example: The Sinking of the Titanic
Do the data support the phrase “Women and children first?”
• 711/2201, or 32% of all passengers survived.
• 367/1731, or 21% of the males survived.
• 344/470, or 73% of the females survived.
• The chance of survival depends on the condition of whether the
passenger was male or female.
The notion that probability can change if you are given
additional information is called conditional probability.
Display 5.39
Survived?
Gender
Male
Female
Total
Yes
367
344
711
No
1364
126
1490
Total
1731
470
2201
Section 5.4 - Conditional Probability
Conditional Probability from the Sample Space
Example: The Titanic and Conditional Probability
Let S be the event that the passenger survived and let F be
the event that the passenger was female. What is the
probability of survival, given that the passenger was female?
344
P(S | F)  P(survived | female) 
 0.732
470
Display 5.39
Survived?
Gender
Male
Female
Total
Yes
367
344
711
No
1364
126
1490
Total
1731
470
2201
Section 5.4 - Conditional Probability
Conditional Probability from the Sample Space
Example: Sampling Without Replacement
When you sample without replacement from a small
population, the probabilities for the second draw depends on
the outcome of the first draw.
Suppose you randomly choose two cards from a standard
deck of 52 cards. Suppose the first card chosen is a heart.
What is the probability that the second card chosen will also
be a a heart?
Section 5.4 - Conditional Probability
Conditional Probability from the Sample Space
Example: Sampling Without Replacement
When you sample without replacement from a small
population, the probabilities for the second draw depends on
the outcome of the first draw.
Draw two cards from a standard deck of 52 cards. Suppose
the first card chosen is a heart. What is the probability that
the second card chosen will also be a heart? After the first
heart is chosen, the sample space is changed. There are
only 51 cards and 12 hearts remaining.
13 1
12
P(Heart) 
  0.250
P(Heart | Heart) 
 0.235
52 4
51
Section 5.4 - Conditional Probability
Conditional Probability from the Sample Space
D18. When you compare sampling with and without replacement, how
does the size of the population affect the comparison? Conditional
probability lets you answer the question quantitatively.
Imagine two populations of students, one large (N = 100) and one small
(N = 4), with half of each population male. Draw random samples of size
n = 2 from each population.
First, consider the small population.
Find P(2nd is M | 1st is M), assuming you sample without
replacement. Then calculate the probability again, this time
with replacement.
Section 5.4 - Conditional Probability
Conditional Probability from the Sample Space
First, consider the small population (N = 4).
Find P(2nd is M | 1st is M), assuming you sample without
replacement. Then calculate the probability again, this time
with replacement.
2 1
P(M )    0.500
4 2
1
Without replacement : P(M | M )   0.333
3
2 1
With replacement : P(M | M )    0.500
4 2
Section 5.4 - Conditional Probability
Conditional Probability from the Sample Space
Next, consider the large population (N = 100).
Find P(2nd is M | 1st is M), assuming you sample without
replacement. Then calculate the probability again, this time
with replacement.
Section 5.4 - Conditional Probability
Conditional Probability from the Sample Space
Next, consider the large population (N = 100).
Find P(2nd is M | 1st is M), assuming you sample without
replacement. Then calculate the probability again, this time
with replacement.
50 1
P(M ) 
  0.500
100 2
49
Without replacement : P(M | M ) 
 0.495
99
50 1
With replacement : P(M | M ) 
  0.500
100 2
Section 5.4 - Conditional Probability
Conditional Probability from the Sample Space
How would you describe the effect of population size on the
difference between the two sampling methods?
Section 5.4 - Conditional Probability
Conditional Probability from the Sample Space
How would you describe the effect of population size on the
difference between the two sampling methods?
If the population size is large relative to the sample size,
sampling without replacement is about the same as
sampling with replacement.
Section 5.4 - Conditional Probability
The Multiplication Rule for P(A and B)
Survived
Female and Survived
Female
P(S|F) = 344 / 470
P(D|F) = 126 / 470
P(F)= 470 / 2201
P(M)= 1731 / 2201
P(F and S) = 344 / 2201
P(F and D) = 126 / 2201
Died
Female and Died
Survived
Male and Survived
P(S|M) = 367/ 1731
P(D|M) = 1364 / 1731
P(M and S) = 367 / 2201
P(M and D) = 1364 / 2201
Male
Died
Male and Died
Section 5.4 - Conditional Probability
The Multiplication Rule for P(A and B)
P(A  B)  P(A)  P(B | A)
P(A  B)  P(B)  P(A | B)
Section 5.4 - Conditional Probability
The Multiplication Rule for P(A and B)
1st Event
2nd Event
P(B|A)
P(A)
A
P(B)
B
P(A|B)
Event
Probability
A and B
P(A) P(B|A)
B and A
P(B) P(A|B)
B
A
Section 5.4 - Conditional Probability
The Multiplication Rule for P(A and B)
Example:
P(F  S)  P(F)  P(S | F)
470 344


2201 470
344

2201
 0.156
Section 5.4 - Conditional Probability
The Definition of Conditional Probability
For any two events A and B such that P(B)  0,
P(A  B)
P(A | B) 
P(B)
Section 5.4 - Conditional Probability
The Definition of Conditional Probability
Example: Rolling Dice
Find the probability that you get a sum of 8, given that you
rolled doubles.
6
P(D) 
36
1
P(8  D) 
36
1
P(8  D) 36 1
P(8 | D) 


6
P(D)
6
36
Section 5.4 - Conditional Probability
Conditional Probability and Medical Tests
Screening tests give an indication of whether a person is likely
to have a particular disease or condition. A two-way table is
often used to show the four possible outcomes of a screening
test:
Test Result
Disease
Positive
Negative
Total
Present
a
b
a+b
Absent
c
d
c+d
Total
a+c
b+d
a+b+c+d
Section 5.4 - Conditional Probability
Conditional Probability and Medical Tests
The effectiveness of screening tests is judged using
conditional probability.
Positive predicted value (PPV )  P(disease | test positive)
a

ac
Negative predicted value (NPV )  P(no disease | test negative)
d

bd
Section 5.4 - Conditional Probability
Conditional Probability and Medical Tests
The effectiveness of screening tests is judged using
conditional probability.
Sensitivity  P(test positive | disease)
a

ab
Specificity  P(test negative | no disease)
d

cd
Section 5.4 - Conditional Probability
Conditional Probability and Medical Tests
A false positive is a positive test result when the patient
does not have the disease or condition.
A false negative is a negative test result when the patient
has the disease or condition.
P( false positive)  P(test positive | no disease)
c

cd
P( false negative)  P(test negative | disease)
b

ab
Section 5.4 - Conditional Probability
E51. A screening test for the detection of a certain disease gives a positive
result 6% of the time for people who do not have the disease. The test
gives a negative result 0.5% of the time for people who do have the
disease. Large scale studies have shown that the disease occurs in about
3% of the population.
Fill in a two-way table showing the results expected for every
100,000 people.
Test Result
Positive
Negative
Total
Yes
Disease
No
Total
100,000
Section 5.4 - Conditional Probability
E51. A screening test for the detection of a certain disease gives a positive
result 6% of the time for people who do not have the disease. The test
gives a negative result 0.5% of the time for people who do have the
disease. Large scale studies have show that the disease occurs in about
3% of the population.
Fill in a two-way table showing the results expected for every
100,000 people.
Test Result
Positive
Negative
Total
Yes
2,500
500
3,000
No
6,000
91,000
97,000
Total
8,500
91,500
100,000
Disease
Section 5.4 - Conditional Probability
What is the probability that a person selected at random tests
positive for this disease?
P(test positive) 
What is the probability that a person selected at random who
tests positive for the disease does not have the disease?
P(no disease | test positive) 
Test Result
Disease
Positive
Negative
Total
Yes
2,500
500
3,000
No
6,000
91,000
97,000
Total
8,500
91,500
100,000
Section 5.4 - Conditional Probability
What is the probability that a person selected at random tests
positive for this disease?
8, 500
P(test positive) 
 0.085
100, 000
What is the probability that a person selected at random who
tests positive for the disease does not have the disease?
6, 000
P(no disease | test positive) 
 0.705
8, 500
Test Result
Positive
Negative
Total
Yes
2,500
500
3,000
No
6,000
91,000
97,000
Total
8,500
91,500
100,000
Disease
Section 5.4 - Conditional Probability
Conditional Probability and Statistical Inference
To calculate a probability, you must work from a model.
For example, what is the probability of observing an even
number of dots on a roll of a die?
The model is that the die is fair.
Using this model, P(even|fair die) = 3/6 = 1/2 = 0.5000.
Suppose you have a die that you suspect isn’t fair. How can
you discredit the model that the die is fair?
Suppose you roll the die 10 times and get an even number
every time. P(10 evens) = (1/2)10 = 0.00098.
The outcome is so unlikely that the model of a fair die should
be rejected.