Lesson 10-3 Conditional Probability TB or not TB (did you get it?) Math-2 Lesson 10-3 Conditional Probability TB or not TB (did you get it?)

Joint and Conditional Probabilities Joint Probability:  p(A and B).  The probability of event A and event B occurring.  It is the probability of the intersection of two or more events.  The probability of the intersection of A and B may be written p(A ∩ B). Example:  the probability that a card is a four and red =p(four and red) = 2/52 = 1/26.   (There are two red fours in a deck of 52, the 4 of hearts and the 4 of diamonds). 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 – and (∩) Conditional Probability – (|)

Joint Probability (overlapping events). Blonde Hair (3) Maria Angelica Bill Jim Amber Girls (3) (1) (2) (2) Girl, not blonde Not girl, blonde Girl, blonde (1) (2) (2)

Conditional Probability Ford Non-Ford total white Not white 7 2 9 4 4 6 7 13 1/9, 2/3,7/9, 0

Joint (overlapping) and Conditional Probabilities Ford Non-Ford total white Not white 7 2 9 4 4 6 7 13

Probability Statements

Tree Diagram Wins Losses Tie Games Total Steelers 7 8 1 16 49ers 10 6 17 14 32 Won/Steeler: 7 Steeler Games: 16 Lost/Steeler: 8 Games: 32 tie/Steeler: 1 Won/49er: 10 49er Games: 16 Lost/49er: 6 tie/49er: 0

What did you notice about how fare “upstream” you go to find numbers for the “marginal” probabilities? Won/Steeler: 7 Steeler Games: 16 Lost/Steeler: 8 Games: 32 tie/Steeler: 1 Won/49er: 10 49er Games: 16 Lost/49er: 6 tie/Steeler: 1

Your turn: 1. Fill in the table. 2. Build a tree diagram and label it. Tails No tails Total Mammals 5 4 Not mammals 7 3 9 10 12 7 19 Tails/mammal: 5 Mammals: 9 Animals 19: no tails/mammal: 4 Tails/not mammal: 7 Not mammals: 10 No tails/not mammal: 3

Writing Probability Statements

Build a tree diagram and label it (without #’s at first). Blue Not Blue Total Ford Chevy 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. Blue Not Blue Total Ford Chevy 15 27 – 15 = 12 27 Blue/Ford: 15 Fords: 27 Cars: Not blue/Ford: 12 Chevy’s: This probability gives you 2 numbers in the table/tree. Blue/Chevy From these 2 numbers you can find a 3rd number. Not Blue/Chevy:

From the probability given, fill in the table or the tree. Blue Not Blue Total Ford Chevy 15 12 27 11 43 - 27 = 16 43 Blue/Ford: 15 Cars: 43 Fords: 27 Not blue/Ford: 12 This probability gives you 2 numbers in the table/tree. Chevy’s: 16 Blue/Chevy 11 You now have enough information to complete the table and the tree. Not Blue/Chevy:

From the probability given, fill in the table or the tree. Blue Not Blue Total Ford Chevy 15 12 27 11 16 – 11 = 5 16 43 Blue/Ford: 15 Cars: 43 Fords: 27 Not blue/Ford: 12 This probability gives you 2 numbers in the table/tree. Chevy’s: 16 Blue/Chevy 11 You now have enough information to complete the table and the tree. Not Blue/Chevy: 5

From the probability given, fill in the table or the tree. Blue Not Blue Total Ford Chevy 15 12 27 11 5 16 15 + 11 = 26 43 Blue/Ford: 15 Cars: 43 Fords: 27 Not blue/Ford: 12 This probability gives you 2 numbers in the table/tree. Chevy’s: 16 Blue/Chevy 11 You now have enough information to complete the table and the tree. 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 Have TB/”+” test: Test Positive: Patients: Don’t have TB/ “+”test: Test Negative: Have TB/ ”neg” test: Don’t have TB/ “neg”test:

From the probability given, fill in the table and the tree. Test Positive Test Negative Total Have TB Don’t have TB 675 725 – 675 = 50 725 Have TB/”+” test: 675 Test Positive: 725 Patients: Don’t have TB/ “+”test: 50 This probability gives you 2 numbers in the table/tree. Test Negative: 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 Test Negative Total Have TB Don’t have TB 675 830 50 1015 – 830 = 185 725 1015 Have TB/”+” test: 675 Test Positive: 725 Patients: 1015 Don’t have TB/ “+”test: 50 This probability gives you 2 numbers in the table/tree. Test Negative: Have TB/ ”neg” test: This provides enough information to file in the rest of the table/tree. Don’t have TB/ “neg”test:

From the probability given, fill in the table and the tree. Test Positive Test Negative Total Have TB Don’t have TB 675 830 – 675 = 155 830 50 185 725 1015 Have TB/”+” test: 675 Test Positive: 725 Patients: 1015 Don’t have TB/ “+”test: 50 This probability gives you 2 numbers in the table/tree. Test Negative: Have TB/ ”neg” test: 155 This provides enough information to file in the rest of the table/tree. Don’t have TB/ “neg”test:

From the probability given, fill in the table and the tree. Test Positive Test Negative Total Have TB Don’t have TB 675 155 830 50 185 725 1015 – 725 = 290 1015 Have TB/”+” test: 675 Test Positive: 725 Patients: 1015 Don’t have TB/ “+”test: 50 This probability gives you 2 numbers in the table/tree. Test Negative: 290 Have TB/ ”neg” test: 155 This provides enough information to file in the rest of the table/tree. Don’t have TB/ “neg”test:

From the probability given, fill in the table and the tree. Test Positive Test Negative Total Have TB Don’t have TB 675 155 830 50 290 – 155 = 135 185 725 290 1015 Have TB/”+” test: 675 Test Positive: 725 Patients: 1015 Don’t have TB/ “+”test: 50 This probability gives you 2 numbers in the table/tree. Test Negative: 290 Have TB/ ”neg” test: 155 This provides enough information to file in the rest of the table/tree. 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: 1000 Tested Negative/ yes TB 19 Tested Positive/no TB: 62 Do NOT Have TB: 620 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