Math-2

Lesson 10-3Conditional 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).

BlondeHair(3)

Girls (3)Bill Jim Amber

MariaAngelica

(2)(1) (2)

Not girl,blonde

(2)

Girl, blonde(1)

Girl, not blonde

(2)

?

?)girl blonde( P

?

?)boy blonde( P

Marginal (conditional) Probability

Ford Non-Ford total

white

Not white

total

2

4

7

0

6

4

13

9

7

?

?)Fordwhite( P

?

?)Ford-notwhite( P

?

?)Fordwhite-not( P

?

?)Ford-notwhite-not( P

Joint (overlapping) andMarginal (conditional) Probabilities

Ford Non-Ford total

white

Not white

total

2

4

7

06

4

13

9

7

?

?)Ford and white( P

?

?)Ford / white-not( P

?

?)Ford and white-not( P

?

?) / whiteFord-not( P

Probability Statements

13

2)Ford and white( P

3

2)Ford| white-not( P

13

4)Ford and white-not( P

9

7)white| Ford-not( P

Tree Diagram

Steeler Games: 16

49erGames: 16

Won/Steeler: 7

Lost/Steeler: 8

Lost/49er: 6

Won/49er: 10

Wins Losses Tie Games Total

Steelers 7 8 1 16

49ers 10 6 0 16

Total 17 14 1 32

Games: 32tie/Steeler: 1

tie/49er: 0

Steeler Games: 16

49erGames: 16

Won/Steeler: 7

Lost/Steeler: 8

Lost/49er: 6

Won/49er: 10

Games: 32tie/Steeler: 1

tie/Steeler: 1

?

?)game 94/( winP

What did you notice about how fare “upstream” you go to find numbers for the “marginal” probabilities?

16

10

Your turn:

Mammals: 9

Not mammals: 10

Tails/mammal: 5

no tails/mammal: 4

No tails/not mammal: 3

Tails/not mammal: 7

Tails No tails Total

Mammals 5 4

Not mammals 7 3

Total

Animals 19:

1. Fill in the table.2. Build a tree diagram and label it.

12 7

9

1019

Writing Probability Statements

?

?)mammal / tail( P

?

?)mammalnot / tailno( P

?

?) tailno / mammal ( P

?

?) / tailmammalnot ( P

?

?)mammal( P

?

?)mammal anot ( P

Fords:

Chevy’s:

Blue/Ford:

Not blue/Ford:

Not Blue/Chevy:

Blue/Chevy

Blue Not Blue Total

Ford

Chevy

Total

Cars:

Build a tree diagram and label it (without #’s at first).

Fords:

Chevy’s:

Blue/Ford:

Not blue/Ford:

Not Blue/Chevy:

Blue/Chevy

Blue Not Blue Total

Ford

Chevy

Total

Cars:

From the probability given, fill in the table or the tree.

27

27

15)car/Ford Blue( P

15 27 – 15 = 12

27

15

This probability gives you 2 numbers in the table/tree.

12

From these 2 numbers you can find a 3rd number.

Fords:

Chevy’s:

Blue/Ford:

Not blue/Ford:

Not Blue/Chevy:

Blue/Chevy

Blue Not Blue Total

Ford

Chevy

Total

Cars:

From the probability given, fill in the table or the tree.

43

27

43 - 27 = 1643

43

11)Chevy and Blue( P

15 12

11

11

This probability gives you 2 numbers in the table/tree.

27

15

12

You now have enough information to complete the table and the tree.

16

Fords:

Chevy’s:

Blue/Ford:

Not blue/Ford:

Not Blue/Chevy:

Blue/Chevy

Blue Not Blue Total

Ford

Chevy

Total

Cars:

From the probability given, fill in the table or the tree.

43

2716

43

43

11)Chevy and Blue( P

15 12

11

11

This probability gives you 2 numbers in the table/tree.

27

15

12

You now have enough information to complete the table and the tree.

16 – 11 = 5

5

16

Fords:

Chevy’s:

Blue/Ford:

Not blue/Ford:

Not Blue/Chevy:

Blue/Chevy

Blue Not Blue Total

Ford

Chevy

Total

Cars:

From the probability given, fill in the table or the tree.

15 + 11 = 26

43

2716

43

43

11)Chevy and Blue( P

15 12

11

11

This probability gives you 2 numbers in the table/tree.

27

15

12

You now have enough information to complete the table and the tree.

5

5

16

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.

Test Positive:

Test Negative:

Have TB/”+” test:

Don’t have TB/ “+”test:

Test Positive Test Negative Total

Have TB

Don’t have TB

Total

Patients:

Build a tree diagram and label it.

Have TB/ ”neg” test:

Don’t have TB/ “neg”test:

Test Positive:

Test Negative:

Have TB/”+” test:

Don’t have TB/ “+”test:

Test Positive Test Negative Total

Have TB

Don’t have TB

Total

Patients:

From the probability given, fill in the table and the tree.

725

725

675) test"/"T( BP

675

Have TB/ ”neg” test:

Don’t have TB/ “neg”test:

725

675

This probability gives you 2 numbers in the table/tree.

From these 2 numbers you can find a 3rd number.

725 – 675 = 50

50

Test Positive:

Test Negative:

Have TB/”+” test:

Don’t have TB/ “+”test:

Test Positive Test Negative Total

Have TB

Don’t have TB

Total

Patients:

From the probability given, fill in the table and the tree.

725

1015

830)T( BP

675

Have TB/ ”neg” test:

Don’t have TB/ “neg”test:

725

675

This probability gives you 2 numbers in the table/tree.

This provides enough information to file in the rest of the table/tree.

50

1015

830

1015

1015 – 830 = 185

50

Test Positive:

Test Negative:

Have TB/”+” test:

Don’t have TB/ “+”test:

Test Positive Test Negative Total

Have TB

Don’t have TB

Total

Patients:

From the probability given, fill in the table and the tree.

725

1015

830)T( BP

675

Have TB/ ”neg” test:

Don’t have TB/ “neg”test:

725

675

This probability gives you 2 numbers in the table/tree.

This provides enough information to file in the rest of the table/tree.

50

1015

830

1015

185

155

830 – 675 = 155

50

Test Positive:

Test Negative:

Have TB/”+” test:

Don’t have TB/ “+”test:

Test Positive Test Negative Total

Have TB

Don’t have TB

Total

Patients:

From the probability given, fill in the table and the tree.

725

1015

830)T( BP

675

Have TB/ ”neg” test:

Don’t have TB/ “neg”test:

725

675

This probability gives you 2 numbers in the table/tree.

This provides enough information to file in the rest of the table/tree.

50

1015

830

1015

185

155

1015 – 725 = 290

290

155

50

Test Positive:

Test Negative:

Have TB/”+” test:

Don’t have TB/ “+”test:

Test Positive Test Negative Total

Have TB

Don’t have TB

Total

Patients:

From the probability given, fill in the table and the tree.

725

1015

830)T( BP

675

Have TB/ ”neg” test:

Don’t have TB/ “neg”test:

725

675

This probability gives you 2 numbers in the table/tree.

This provides enough information to file in the rest of the table/tree.

50

1015

830

1015

185

155

290

290

290 – 155 = 135155

135

50

Below is a tree diagram representing data based on 1,000 people who have been given a skin test for tuberculosis.

Have TB: 380

Do NOTHave TB: 620

Tested Positive/yes TB: 361

Tested Negative/ yes TB 19

Tested Negative/no TB: 553

Tested Positive/no TB: 62

# tested: 1000

Homework 10.3

• Finish the TB Activity• Part 1: Fill in table, Questions 1-2• Part 2: Questions 1-7