Math-2
Lesson 10-3
Conditional Probability
TB or not TB (did you get it?)
Marginal and Conditional Probability
• Marginal probability: the probability of an event occurring
(p(A)), it may be thought of as an unconditional
probability. It is not conditioned on another event.
– Example: the probability that a card drawn is red (p(red) =
0.5). Another example: the probability that a card drawn is a
4 (p(four)=1/13).
• Conditional probability: p(A|B) is the probability of event
A occurring, given that event B occurs.
– Example: given that you drew a red card, what’s the probability
that it’s a four (p(four|red))=2/26=1/13. So out of the 26 red
cards (given a red card), there are two fours so 2/26=1/13.
Review of Probabilities
• Joint (overlapping) Probability
• Marginal (conditional) Probability
Joint Probability (overlapping events).
Blonde
Hair
(3)
Bill
Jim
(2)
Amber
(1)
Maria
Angelica
Girls (3)
(2)
?
P (blonde girl ) 
?
?
P (blonde boy) 
?
Not girl,
blonde
(2)
Girl, blonde
(1)
Girl,
not blonde
(2)
Marginal (conditional) Probability
Ford
Non-Ford
total
white
2
7
9
Not white
4
0
4
total
6
7
13
?
P( white Ford) 
?
?
P( white not - Ford ) 
?
?
P(not - white Ford ) 
?
?
P (not - white not - Ford ) 
?
Joint (overlapping) and
Marginal (conditional) Probabilities
Ford
white
Not white
total
2
4
6
?
P( white and Ford) 
?
?
P(not - white / Ford) 
?
Non-Ford
7
0
7
total
9
4
13
?
P(not - white and Ford ) 
?
?
P(not - Ford / white ) 
?
Probability Statements
2
P ( white and Ford ) 
13
2
P(not - white | Ford) 
3
4
P(not - white and Ford ) 
13
7
P(not - Ford | white ) 
9
Tree Diagram
Wins
Losses
Tie Games
Total
Steelers
7
8
1
16
49ers
10
6
0
16
Total
17
14
1
32
Steeler
Games: 16
Games: 32
Won/Steeler: 7
Lost/Steeler: 8
tie/Steeler: 1
49er
Games: 16
Won/49er: 10
Lost/49er: 6
tie/49er: 0
What did you notice about how fare “upstream” you
go to find numbers for the “marginal” probabilities?
?
P( win / 49 game ) 
?
Steeler
Games: 16
Games: 32
10

16
Won/Steeler: 7
Lost/Steeler: 8
tie/Steeler: 1
49er
Games: 16
Won/49er: 10
Lost/49er: 6
tie/Steeler: 1
1. Fill in the table.
2. Build a tree diagram and label it.
Your turn:
Tails
No tails
Total
Mammals
5
4
9
Not mammals
7
3
Total
12
7
10
19
Tails/mammal: 5
Mammals: 9
Animals 19:
no tails/mammal: 4
Not
mammals: 10
Tails/not mammal: 7
No tails/not mammal: 3
Writing Probability Statements
?
P(mammal ) 
?
?
P( tail / mammal ) 
?
?
P( mammal / no tail ) 
?
?
P(not a mammal ) 
?
?
P(no tail / not mammal ) 
?
?
P ( not mammal / tail ) 
?
Build a tree diagram and label it (without #’s at first).
Blue
Not Blue
Total
Ford
Chevy
Total
Blue/Ford:
Fords:
Cars:
Not blue/Ford:
Chevy’s:
Blue/Chevy
Not Blue/Chevy:
From the probability given, fill in the table or the tree.
Ford
Blue
Not Blue
Total
15
27 – 15 = 12
27
Chevy
Total
15
P(Blue car/Ford ) 
27
Blue/Ford: 15
Fords: 27
Cars:
This probability gives you 2
numbers in the table/tree.
From these 2 numbers you
can find a 3rd number.
Not blue/Ford: 12
Chevy’s:
Blue/Chevy
Not Blue/Chevy:
From the probability given, fill in the table or the tree.
Ford
Chevy
Blue
Not Blue
Total
15
11
12
27
43 - 27 = 16
43
Total
Blue/Ford: 15
11
P(Blue and Chevy ) 
43
Cars: 43
This probability gives you 2
numbers in the table/tree.
You now have enough
information to complete
the table and the tree.
Fords:
27
Chevy’s: 16
Not blue/Ford: 12
Blue/Chevy 11
Not Blue/Chevy:
From the probability given, fill in the table or the tree.
Ford
Chevy
Blue
Not Blue
Total
15
11
12
27
16
16 – 11 = 5
43
Total
Blue/Ford: 15
11
P(Blue and Chevy ) 
43
Cars: 43
This probability gives you 2
numbers in the table/tree.
You now have enough
information to complete
the table and the tree.
Fords:
27
Chevy’s: 16
Not blue/Ford: 12
Blue/Chevy 11
Not Blue/Chevy: 5
From the probability given, fill in the table or the tree.
Ford
Chevy
Total
Blue
Not Blue
Total
15
11
12
27
16
5
15 + 11 = 26
43
Blue/Ford: 15
11
P(Blue and Chevy ) 
43
Cars: 43
This probability gives you 2
numbers in the table/tree.
You now have enough
information to complete
the table and the tree.
Fords:
27
Chevy’s: 16
Not blue/Ford: 12
Blue/Chevy 11
Not Blue/Chevy: 5
TB or Not TB?
Tuberculosis (TB) can be tested in a
variety of ways, including a skin test.
If a person has tuberculosis
antibodies, then they are considered
to have TB.
Build a tree diagram and label it.
Test Positive
Test Negative
Total
Have TB
Don’t have TB
Total
Have TB/”+” test:
Patients:
Test Positive:
Test Negative:
Don’t have TB/ “+”test:
Have TB/ ”neg” test:
Don’t have TB/ “neg”test:
From the probability given, fill in the table and the tree.
Test Positive
Have TB
Don’t have TB
Test Negative
Total
675
725 – 675 = 50
Total
725
Have TB/”+” test: 675
675
P(TB/"" test ) 
725
Patients:
This probability gives you 2
numbers in the table/tree.
Test Positive: 725
Test Negative:
Don’t have TB/ “+”test: 50
Have TB/ ”neg” test:
Don’t have TB/ “neg”test:
From these 2 numbers you
can find a 3rd number.
From the probability given, fill in the table and the tree.
Test Positive
Have TB
Don’t have TB
675
50
Total
725
Test Negative
830
1015 – 830 = 185
1015
Have TB/”+” test: 675
830
P (TB ) 
1015
Patients: 1015
This probability gives you 2
numbers in the table/tree.
This provides enough
information to file in the
rest of the table/tree.
Total
Test Positive: 725
Test Negative:
Don’t have TB/ “+”test: 50
Have TB/ ”neg” test:
Don’t have TB/ “neg”test:
From the probability given, fill in the table and the tree.
Test Positive
Have TB
Don’t have TB
675
50
Total
725
Test Negative
Total
830 – 675 = 155
830
185
1015
Have TB/”+” test: 675
830
P (TB ) 
1015
Patients: 1015
This probability gives you 2
numbers in the table/tree.
This provides enough
information to file in the
rest of the table/tree.
Test Positive: 725
Test Negative:
Don’t have TB/ “+”test: 50
Have TB/ ”neg” test: 155
Don’t have TB/ “neg”test:
From the probability given, fill in the table and the tree.
Test Positive
Have TB
Don’t have TB
675
50
Total
725
Test Negative
Total
155
830
185
1015 – 725 = 290
1015
Have TB/”+” test: 675
830
P (TB ) 
1015
Patients: 1015
This probability gives you 2
numbers in the table/tree.
This provides enough
information to file in the
rest of the table/tree.
Test Positive: 725
Don’t have TB/ “+”test: 50
Test Negative: 290
Have TB/ ”neg” test: 155
Don’t have TB/ “neg”test:
From the probability given, fill in the table and the tree.
Test Positive
Have TB
675
Don’t have TB
50
Total
725
Test Negative
Total
155
290 – 155 = 135
830
185
290
1015
Have TB/”+” test: 675
830
P (TB ) 
1015
Patients: 1015
This probability gives you 2
numbers in the table/tree.
This provides enough
information to file in the
rest of the table/tree.
Test Positive: 725
Don’t have TB/ “+”test: 50
Test Negative: 290
Have TB/ ”neg” test: 155
Don’t have TB/ “neg”test: 135
Below is a tree diagram representing data based on
1,000 people who have been given a skin test for
tuberculosis.
Tested Positive/yes TB: 361
Have TB: 380
Tested Negative/ yes TB 19
# tested: 1000
Do NOT
Have TB: 620
Tested Positive/no TB: 62
Tested Negative/no TB: 553
Homework 10.3
• Finish the TB Activity
• Part 1: Fill in table, Questions 1-2
• Part 2: Questions 1-7