# Section 6.3 Probability Models Statistics AP Mrs. Skaff.

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Section 6.3 Probability Models Statistics AP Mrs. Skaff

Today you will learn how to… Construct Venn Diagrams, Tables, and Tree Diagrams and use them to calculate probabilities Calculate probabilities for nondisjoint events Modify the multiplication rule to accommodate non-independent events Calculate conditional probabilities AP Statistics, Section 6.3, Part 12

Non-Independent Events AP Statistics, Section 6.3, Part 13 You draw a card from a deck and then draw another one without replacing the first card. What is the probability that you draw a diamond and then another diamond?

GENERAL MULTIPLICATION RULE The joint probability that both of two events A and B happen together can be found by P(A and B) = P(A)P(B|A) Here P(B|A) is the conditional probability that B occurs given the information that A occurs. 4

5 Venn Diagrams: Disjoint Events A B S

6 A B S Rule #3 (addition rule for disjoint events!) P(A or B) = P(A) + P(B)

7 Venn Diagrams: Non-disjoint Events A B S A and B P(A or B) = P(A) + P(B) – P(A and B)

8 Example Claire and Alex are awaiting the decision about a promotion. Claire guesses her probability of her getting a promotion at.7 and Alex’s probability at.5. Claire also thinks the probability of both getting promoted is.3

9 Example What’s the probability of either Claire or Alex getting promoted P(C or A)?  P(C) = 0.7  P(A) = 0.5  P(C and A) = 0.3

10 Example What’s the probability of either Claire or Alex getting promoted P(C or A)? P(C) = 0.7 P(A) = 0.5 P(C and A) = 0.3 A C A and C

11 Example What’s the probability of either Claire or Alex getting promoted P(C or A)? A 0.2 C 0.4 A and C 0.3 0.1

12 Example P(C and A c )? P(A and C c )? P(A c and C c )? A 0.2 C 0.4 A and C 0.3 0.1

13 Example Are the events, Alex gets a promotion and Claire gets a promotion independent? Why? A 0.2 C 0.4 A and C 0.3 0.1

Charts! Woohoo! Charts are AWESOME! AP Statistics, Section 6.3, Part 114

15 Age 18-2930-6465 and overTotal Married7,84243,8088,27059,920 Never Married 13,9307,18475121,865 Widowed362,5238,38510,944 Divorced7049,1741,26311,141 Total22,51262,68918,669103,870

16 Age 18-2930-6465 and overTotal Married7,84243,8088,27059,920 Never Married 13,9307,18475121,865 Widowed362,5238,38510,944 Divorced7049,1741,26311,141 Total22,51262,68918,669103,870 A=is young (between 18 and 29) P(A)=

17 Age 18-2930-6465 and overTotal Married7,84243,8088,27059,920 Never Married 13,9307,18475121,865 Widowed362,5238,38510,944 Divorced7049,1741,26311,141 Total22,51262,68918,669103,870 B=married P(B)=

18 Age 18-2930-6465 and overTotal Married7,84243,8088,27059,920 Never Married13,9307,18475121,865 Widowed362,5238,38510,944 Divorced7049,1741,26311,141 Total22,51262,68918,669103,870 A=is young (between 18 and 29) B=married P(A and B)=

19 Age 18-2930-6465 and overTotal Married7,84243,8088,27059,920 Never Married13,9307,18475121,865 Widowed362,5238,38510,944 Divorced7049,1741,26311,141 Total22,51262,68918,669103,870 A=is young (between 18 and 29) B=married P(A | B)= (Read as “the probability of A given B”)  The probability that a a person is young, given that they are married This is known as a “conditional probability”

20 Age 18-2930-6465 and overTotal Married7,84243,8088,27059,920 Never Married13,9307,18475121,865 Widowed362,5238,38510,944 Divorced7049,1741,26311,141 Total22,51262,68918,669103,870 A=is young (between 18 and 29) B=married P(A | B)= (Read as “the probability of A given B”)  The probability that a a person is young, given that they are married This is known as a “conditional probability”

Conditional Probabilities Conditional probability measures the probability of an event A occurring given that B has already occurred. There is a formula for this in your packets. Sometimes it can be confusing for solving real-life problems… It is usually easier to use a tree diagram, venn diagram, or table to solve these problems!!! AP Statistics, Section 6.3, Part 121

22 Age 18-2930-6465 and overTotal Married7,84243,8088,27059,920 Never Married13,9307,18475121,865 Widowed362,5238,38510,944 Divorced7049,1741,26311,141 Total22,51262,68918,669103,870 A=is young (between 18 and 29) B=married P(A | B)= P(B | A) =

Conditional Probabilities and Tables Bag A contains 5 blue and 4 green marbles. Bag B contains 3 yellow, 4 blue, and 2 green marbles. Given you have a green marble, what is the probability it came from Bag A? AP Statistics, Section 6.3, Part 123

24 Conditional Probabilities and Venn Diagrams What is the probability of Claire being promoted given that Alex got promoted? A 0.2 C 0.4 A and C 0.3 0.1

Probabilities with Tree Diagrams Example: A videocassette recorder (VCR) manufacturer receives 70% of his parts from factory F1 and the rest from factory F2. Suppose 3% of the output from F1 are defective, while only 2% of the output from F2 are defective. What is the probability the part is defective? 25 F1 F2 G D G D F1 Good F1 Defective F2 Good F2 Defective 0.7 0.3 Example: A videocassette recorder (VCR) manufacturer receives 70% of his parts from factory F1 and the rest from factory F2. Suppose 3% of the output from F1 are defective, while only 2% of the output from F2 are defective. What is the probability the part is defective? 0.97 0.03 Example: A videocassette recorder (VCR) manufacturer receives 70% of his parts from factory F1 and the rest from factory F2. Suppose 3% of the output from F1 are defective, while only 2% of the output from F2 are defective. What is the probability the part is defective? 0.98 0.02 0.679 0.021 0.294 0.006 Example: A videocassette recorder (VCR) manufacturer receives 70% of his parts from factory F1 and the rest from factory F2. Suppose 3% of the output from F1 are defective, while only 2% of the output from F2 are defective. What is the probability the part is defective?

Given that a randomly chosen part is defective, what is the probability that it came from factory F1? 26 F1 F2 G D G D F1 Good F1 Defective F2 Good F2 Defective 0.7 0.3 0.97 0.03 0.98 0.02 0.679 0.021 0.294 0.006

Summary…You should be able to: AP Statistics, Section 6.3, Part 127 Construct a Venn Diagram and use it to calculate probabilities  Particularly useful for nondisjoint events Fill in a table of probabilities and use it to calculate probabilities  Especially useful for conditional probabilities Construct a Tree Diagram and use it to calculate probabilities

Summary…Formulas AP Statistics, Section 6.3, Part 128 Probability of nondisjoint events Multiplication Rule P(A and B) = P(A)P(B|A) Conditional Probabilities P(A | B)= P(A and B) / P(B)

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