Probability of Independent and Dependent Events and Review

1. Probability of Independent and Dependent Events and Review Probability & Statistics 1.0 Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces. 2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.

2. Probability of Independent and Dependent Events and Review Objectives Key Words Independent Events The occurrence of one event does not affect the occurrence of the other Dependents Events The occurrence of one event doesaffect the occurrence of the other Conditional Probability Two dependent events A and B, the probability that B will occur given that A has occurred. • Solve for the probability of an independent event. • Solve for the probability of a dependent event.

3. Example 1 Tell whether the events are independent or dependent. Explain. a. Your teacher chooses students at random to present their projects. She chooses you first, and then chooses Kim from the remaining students. b. You flip a coin, and it shows heads. You flip the coin again, and it shows tails. c. One out of 25 of a model of digital camera has some random defect. You and a friend each buy one of the cameras. You each receive a defective camera. Identify Events

4. Example 1 SOLUTION a. Dependent; after you are chosen, there is one fewer student from which to make the second choice. b. Independent; what happens on the first flip has no effect on the second flip. c. Independent; because the defects are random, whether one of you receives a defective camera has no effect on whether the other person does too. Identify Events

5. Checkpoint 1. Tell whether the events are independent or dependent. Explain. You choose Alberto to be your lab partner. Then Tia chooses Shelby. ANSWER dependent 2. You spin a spinner for a board game, and then you roll a die. ANSWER independent Identify Events

6. Example 2 Concerts A high school has a total of 850 students. The table shows the numbers of students by grade at the school who attended a concert. a. What is the probability that a student at the school attended the concert? b. What is the probability that a junior did not attend the concert? Find Conditional Probabilities Did not attend Grade Attended Freshman 80 120 Sophomore 132 86 Junior 173 29 Senior 179 51

7. Example 2 juniors who did not attend b. P(did not attend junior) = total juniors 29 0.144 = = 173 29 + ~ ~ ~ ~ SOLUTION total who attended 80 132 173 179 + + + a. P(attended) = = 850 total students 564 29 0.664 = 850 202 Find Conditional Probabilities

8. Checkpoint 3. Use the table below to find the probability that a student is a junior given that the student did not attend the concert. Did not attend 0.101 Grade Attended ANSWER Freshman 80 120 Sophomore 132 86 Junior 173 29 ~ ~ Senior 179 51 29 286 Find Conditional Probabilities

9. Probability of Independent and Dependent Events • Independent Events • If A and B are independent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B) • Dependent Events • If A and B are dependent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B|A)

10. Example 3 Independent and Dependent Events Games A word game has 100 tiles, 98 of which are letters and two of which are blank. The numbers of tiles of each letter are shown in the diagram. Suppose you draw two tiles. Find the probability that both tiles are vowels in the situation described. a. You replace the first tile before drawing the second tile. b. You do not replace the first tile before drawing the second tile.

11. Example 3 42 42 ( ( ( A and B P ( P A ( P B ( 0.1764 • • = = = 100 100 Independent and Dependent Events SOLUTION a. If you replace the first tile before selecting the second, the events are independent. Let A represent the first tile being a vowel and B represent the second tile being a vowel. Of 100 tiles, 9 12 9 8 4 42 + + + + = are vowels.

12. Example 3 b. If you do not replace the first tile before selecting the second, the events are dependent. After removing the first vowel, 41 vowels remain out of 99 tiles. 42 41 ~ ( ( ~ | P A ( P B A ( 0.1739 • • = = 100 99 ( A and B P ( Independent and Dependent Events

13. Checkpoint 1 0.0001; ANSWER = 10,000 1 ~ ~ 0.0002 4950 Find Probabilities of Independent and Dependent Events 4. In the game in Example 3, you draw two tiles. What is the probability that you draw a Q, then draw a Z if you first replace the Q? What is the probability that you draw both of the blank tiles (without replacement)?

14. Conclusion Summary Assignment Probability of Independent and Dependent Events Page 572 #(11-14,15,18,22,26,30) • How are probabilities calculated for two events when the outcome of the first event influences the outcome of the second event? • Multiply the probability of the second event, given that the first event happen.

15. Review Probability & Statistics 1.0 Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces. 2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.

16. Number of outcomes in event 3 P(red) = = Total number of outcomes 8 Theoretical Probability of an Event When all outcomes are equally likely, the theoretical probability that an event A will occur is: The theoretical probability of an event is often simply called its probability. Example: What is the probability that the spinner shown lands on red if it is equally likely to land on any section? Solution: The 8 sections represent the 8 possible outcomes. Three outcomes correspond to the event “lands on red.”

17. Number preferring sneakers P(prefers sneakers) = Total number of students 820 0.48 = 1700 ~ ~ ~ ~ Number preferring shoes or boots P(prefers shoes or boots) = Total number of students 340 0.19 = 1700 Experimental Probability of an Event For a given number of trials of an experiment, the experimental probability that an event A will occur is: Solution: Find the total number of students surveyed. 820+556+204+120=1700 a. Of 1700 students, 820 prefer sneakers. b. Of 1700 students surveyed, prefer shoes or boots. Example: SurveysThe graph shows results of a survey asking students to name their favorite type of footwear. What is the experimental probability that a randomly chosen student prefers Sneakers? Shoes or boots?

18. Probability of Compound Events • Overlapping Events • If A and B are overlapping events, then P(A and B)≠0, and the probability of A or B is: • Disjoint Events • If A and B are disjoint events, then P(A and B)=0, and the probability of A or B is:

19. Probability of the Complement of an Event The sum of the probabilities of an event and its complement is 1. So, Recall: Complement of an Event All outcomes that are not in the event

20. Probability of Independent and Dependent Events • Independent Events • If A and B are independent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B) • Dependent Events • If A and B are dependent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B|A)