1. Introduction In probability, events are either dependent or independent. Two events are independent if the occurrence or non-occurrence of one event has no effect on the probability of the other event. If two events are independent, then you can simply multiply their individual probabilities to find the probability that both events will occur. If events are dependent, then the outcome of one event affects the outcome of another event. So it is important to know whether or not two events are independent. 1 7.1.3: Understanding Independent Events

2. Introduction, continued Two events A and B are independent if and only if P(A and B) = P(A) P(B). But sometimes you only know two of these three probabilities and you want to find the third. In such cases, you can’t test for independence, so you might assess the situation or nature of the experiment and then make an assumption about whether or not the events are independent. Then, based on your assumption, you can find the third probability in the equation. 2 7.1.3: Understanding Independent Events

3. Key Concepts Two events A and B are independent if and only if they satisfy the following test: P(A and B) = P(A) P(B) Using set notation, the test is Sometimes it is useful or necessary to make an assumption about whether or not events are independent, based on the situation or nature of the experiment. Then any conclusions or answers are based on the assumption. 3 7.1.3: Understanding Independent Events

4. Key Concepts, continued In a uniform probability model, all the outcomes of an experiment are assumed to be equally likely, and the probability of an event E, denoted P(E), is given by When this definition of probability is used and the relevant probabilities are known, then the definition of independent events can be used to determine, verify, or prove that two events are dependent or independent. 4 7.1.3: Understanding Independent Events

5. Key Concepts, continued The relative frequency of an event is the number of times it occurs divided by the number of times the experiment is performed (called trials) or the number of observations: Relative frequency can be used to estimate probability in some cases where a uniform probability model does not seem appropriate. Such cases include data collected by surveys or obtained by observations. 5 7.1.3: Understanding Independent Events

6. Key Concepts, continued When relative frequency is used to estimate the relevant probabilities, then the definition of independent events can be used to determine whether two events seem to be dependent or independent, based on the data. Probability and relative frequency are related as follows: 6 7.1.3: Understanding Independent Events

7. Key Concepts, continued The probability of an event can be used to predict its relative frequency if the experiment is performed a large number of times. For example, the probability of getting a 3 by rolling a fair die is. So if you roll a fair die 6,000 times, it is reasonable to predict that you will get a 3 about 1,000 times, or of the number of times you roll the die. If you roll a die 6,000 times, the number of times you get a 3 might not be 1,000. 7 7.1.3: Understanding Independent Events

8. Key Concepts, continued Relative frequency can be used to predict the probability of an event. In general, as the number of trials or observations increases, the prediction becomes stronger. For example, suppose you ask 40 people for their favorite ice cream flavor. If 8 say chocolate, then you can predict that the probability of a randomly selected person saying chocolate is, or 20%. But 40 people make up a small sample, so this is not a very strong prediction. 8 7.1.3: Understanding Independent Events

9. Key Concepts, continued Now suppose you ask 4,000 people who are randomly selected using a good sampling method. If 740 say chocolate, then you can predict that the probability of a randomly selected person saying chocolate is, or 18.5%; this is a stronger prediction. 9 7.1.3: Understanding Independent Events

10. Key Concepts, continued Also remember the Addition Rule: If A and B are any two events, then the probability of A or B, denoted P(A or B), is given by P(A or B) = P(A) + P(B) – P(A and B). Using set notation, the rule is 10 7.1.3: Understanding Independent Events

11. Common Errors/Misconceptions thinking that the actual relative frequency of an event will equal the probability of the event; for example, thinking 100 tosses of a fair coin will yield 50 heads and 50 tails thinking that a probability based on actual relative frequency is a “true fact;” that is, not understanding that the probability is just a number that is only as good as the assumptions and statistical sampling methods it is based on 11 7.1.3: Understanding Independent Events

12. Common Errors/Misconceptions, continued thinking that a probability based on past actual relative frequency and obtained using reasonable assumptions and sound statistical sampling methods can be used to guarantee the actual future relative frequency of an event; that is, not realizing that even a probability obtained by the best possible means is only a predictor, never a guarantee 12 7.1.3: Understanding Independent Events

13. Guided Practice Example 2 Trevor tosses a coin 3 times. Consider the following events. For each of the following pairs of events, determine if the events are independent. 13 7.1.3: Understanding Independent Events

14. Guided Practice: Example 2, continued 1.List the sample space. Sample space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} 14 7.1.3: Understanding Independent Events

15. Guided Practice: Example 2, continued 2.Use the sample space to determine the relevant probabilities. 15 7.1.3: Understanding Independent Events There are 4 outcomes with heads first. There are 4 outcomes with heads second. There are 2 outcomes with exactly 2 consecutive heads.

16. Guided Practice: Example 2, continued 16 7.1.3: Understanding Independent Events There are 2 outcomes with heads first and heads second. There is 1 outcome with heads first and exactly 2 consecutive heads. There are 2 outcomes with heads second and exactly 2 consecutive heads.

17. Guided Practice: Example 2, continued 3.Use the definition of independence to determine if the events are independent in each specified pair. A and B are independent. 17 7.1.3: Understanding Independent Events

18. Guided Practice: Example 2, continued A and C are independent. B and C are dependent. 18 7.1.3: Understanding Independent Events ✔

19. Guided Practice: Example 2, continued 19 7.1.3: Understanding Independent Events

20. Guided Practice Example 3 Landen owns a delicatessen. He collected data on sales of his most popular sandwiches for one week and recorded it in the table below. 20 7.1.3: Understanding Independent Events Bread choice Sandwich choice Landen’s club Turkey melt Roasted chicken Veggie delight Country white4425 8 Whole wheat24282634 Sourdough24272431

21. Guided Practice Example 3, continued Each of the following statements describes a pair of events. For each statement, determine if the events seem to be independent based on the data in the table. A random customer orders Landen’s club sandwich on country white bread. A random customer orders the roasted chicken sandwich on whole wheat bread. 21 7.1.3: Understanding Independent Events

22. Guided Practice: Example 3, continued 1.Find the totals of all the categories. 22 7.1.3: Understanding Independent Events Bread choice Sandwich choice Total Landen’s club Turkey melt Roasted chicken Veggie delight Country white 4425 8102 Whole wheat 24282634112 Sourdough 24272431106 Total 92807573320

23. Guided Practice: Example 3, continued 2.For the first statement, assign variables as names of the events. LC: A random customer orders Landen’s club sandwich. CW: A random customer orders country white bread. 23 7.1.3: Understanding Independent Events

24. Guided Practice: Example 3, continued 3.Use the data to determine the relevant probabilities. 24 7.1.3: Understanding Independent Events There were 320 sandwiches sold, and 92 of them were Landen’s club. There were 320 sandwiches sold, and 102 of them were on country white bread. There were 320 sandwiches sold, and 44 of them were Landen’s club on country white bread.

25. Guided Practice: Example 3, continued 4.Use the definition of independence to determine if the events seem to be independent. 25 7.1.3: Understanding Independent Events Substitute probabilities and simplify. LC and CW seem to be dependent, based on the data.

26. Guided Practice: Example 3, continued 5.For the second statement, assign variables as names of the events. RC: A random customer orders a roasted chicken sandwich. WW: A random customer orders whole wheat bread. 26 7.1.3: Understanding Independent Events

27. Guided Practice: Example 3, continued 6.Use the data to determine the relevant probabilities. 27 7.1.3: Understanding Independent Events There were 320 sandwiches sold, and 75 of them were roasted chicken. There were 320 sandwiches sold, and 112 of them were on whole wheat bread. There were 320 sandwiches sold, and 26 of them were roasted chicken on whole wheat bread.

28. Guided Practice: Example 3, continued 7.Use the definition of independence to determine if the events seem to be independent. RC and WW seem to be independent based on the data. 28 7.1.3: Understanding Independent Events Substitute probabilities and simplify. ✔

29. Guided Practice: Example 3, continued 29 7.1.3: Understanding Independent Events