1. 7/20 The following table shows the number of people that like a particular fast food restaurant. 1)What is the probability that a person likes Wendy’s? 2)What is the probability that a person is male given they like Burger King? 3. What is the probability that a randomly chosen person is female or likes McDonald’s? 3/5 McDonald’sBurger KingWendy’s Male201510 Female201025 3/4

2. EOCT Review

3. CCGPS Geometry UNIT QUESTION: What connection does conditional probability have to independence? Standard: MCC9-12.S.CP.1-7 Today’s Question: How can I determine if 2 events are independent of each other? Standard: MCC9-12.S.CP.1, 7

4. Probability Independent vs. Dependent events

5. Independent Events Two events A and B, are independent if the fact that A occurs does not affect the probability of B occurring. Examples- Landing on heads from two different coins, rolling a 4 on a die, then rolling a 3 on a second roll of the die. Probability of A and B occurring: P(A and B) = P(A)  P(B)

6. Experiment 1 A coin is tossed and a 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die. P (head)=1/2 P (head)=1/2 P(3)=1/6 P(3)=1/6 P (head and 3)=P (head)  P(3) P (head and 3)=P (head)  P(3) =1/2  1/6 =1/2  1/6 = 1/12 = 1/12

7. Experiment 2 A card is chosen at random from a deck of 52 cards. It is then replaced and a second card is chosen. What is the probability of choosing a jack and an eight? P (jack)= 4/52 P (jack)= 4/52 P (8)= 4/52 P (8)= 4/52 P (jack and 8)= 4/52  4/52 P (jack and 8)= 4/52  4/52 = 1/169 = 1/169

8. Experiment 3 A jar contains three red, five green, two blue and six yellow marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a green and a yellow marble? P (green) = 5/16 P (green) = 5/16 P (yellow) = 6/16 P (yellow) = 6/16 P (green and yellow) = P (green)  P (yellow) P (green and yellow) = P (green)  P (yellow) = 15 / 128 = 15 / 128

9. Experiment 4 A school survey found that 9 out of 10 students like pizza. If three students are chosen at random with replacement, what is the probability that all three students like pizza? P (student 1 likes pizza) = 9/10 P (student 1 likes pizza) = 9/10 P (student 2 likes pizza) = 9/10 P (student 2 likes pizza) = 9/10 P (student 3 likes pizza) = 9/10 P (student 3 likes pizza) = 9/10 P (student 1 and student 2 and student 3 like pizza) = 9/10  9/10  9/10 = 729/1000 P (student 1 and student 2 and student 3 like pizza) = 9/10  9/10  9/10 = 729/1000

10. Dependent Events Two events A and B, are dependent if the fact that A occurs affects the probability of B occurring. Examples- Picking a blue marble and then picking another blue marble if I don’t replace the first one. Probability of A and B occurring: P(A and B) = P(A)  P(B|A)

11. Experiment 1 A jar contains three red, five green, two blue and six yellow marbles. A marble is chosen at random from the jar. A second marble is chosen without replacing the first one. What is the probability of choosing a green and a yellow marble? P (green) = 5/16 P (yellow given green) = 6/15 P (green and then yellow) = P (green)  P (yellow) = 1/8 = 1/8

12. Experiment 2 An aquarium contains 6 male goldfish and 4 female goldfish. You randomly select a fish from the tank, do not replace it, and then randomly select a second fish. What is the probability that both fish are male? P (male) = 6/10 P (male) = 6/10 P (male given 1 st male) = 5/9 P (male given 1 st male) = 5/9 P (male and then, male) = 1/3 P (male and then, male) = 1/3

13. Experiment 3 A random sample of parts coming off a machine is done by an inspector. He found that 5 out of 100 parts are bad on average. If he were to do a new sample, what is the probability that he picks a bad part and then, picks another bad part if he doesn’t replace the first? P (bad) = 5/100 P (bad) = 5/100 P (bad given 1 st bad) = 4/99 P (bad given 1 st bad) = 4/99 P (bad and then, bad) = 1/495 P (bad and then, bad) = 1/495

14. Example There are 120 M&M’s in a jar. 25 are blue, 24 are red, 30 are green, 26 are yellow, and 15 are brown. What is the probability of choosing one red, one green and one yellow M&M if choosing three and replacing after each choice??? How does the probability change if there is no replacement? What if we’re trying to choose three green? How does replacement affect that probability?

15. Independent vs. Dependent Determining if 2 events are independent

16. Independent Events Two events are independent if the following are true: P(A|B) = P(A) P(B|A) = P(B) P(A AND B) = P(A) ⋅ P(B) To show 2 events are independent, you must prove one of the above conditions.

17. Experiment 1 Let event G = taking a math class. Let event H = taking a science class. Then, G AND H = taking a math class and a science class. Suppose P(G)=0.6, P(H)=0.5, and P(G AND H)=0.3. Are G and H independent?

18. Experiment 2 In a particular college class, 60% of the students are female. 50% of all students in the class have long hair. 45% of the students are female and have long hair. Of the female students, 75% have long hair. Let F be the event that the student is female. Let L be the event that the student has long hair. One student is picked randomly. Are the events of being female and having long hair independent?

19. Approach #2 If they are independent, P(L|F) should equal P(L). 0.75 ≠ 0.5

20. Homework Practice Worksheet