Unit 13.3A.notebook May 06, 2015 Think write-Pair-share a) Can a subset of A contain elements of A c ? Why or why not? SCP.2 Unit 13. 3 Statistics: Independence of two events with Product Rule b) When you calculate the theoretical probabilities of events based on the same probability experiment, can the term in the denominator of the probability ratio change? Explain. Objective Students will understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities. c) If a set has no elements, what is the probability of the event represented by the set? Explain. d) If the elements of a set are the same as the elements of the sample space, what is the probability of the event represented by the set? Explain. e) Why must you subtract P (A and B) from P (A) + P (B) to determine P (A or B)? Vocabulary Independent Events : when the occurrence of one event has no effect on the occurrence of another event, the two events Think write-Pair-share What does it mean for two events to be independent? Example Suppose you flip a coin and roll a number cube. You would expect the probability of getting heads on the coin to be 1/2 regardless of what number you get from rolling the number cube so events are independent. Likewise, you would expect the probability of rolling a 3 on the number cube to be 1/6 regardless of whether of the coin flip results in heads or tails so the events are independent. Give two examples of independent events REPLACEMENT Because the item is replaced, it resets the event back to the original arrangement and no probabilities are altered. Thus REPLACEMENT tells us that the events are INDEPENDENT NO REPLACEMENT Because the item is NOT replaced, the probabilities are altered. Thus NO REPLACEMENT tells us that the events are NOT INDEPENDENT. Example 1 Example 2 What is the probability of rolling a 6 and then rolling a 5? Given a bag of marbles with 3 red, 2 green and 5 yellow. What is the probability of choosing a red, replacing it and then choosing a green? Are these event independent? why? Are these event independent? why? Unit 13.3A.notebook We can use the formal relationship of P (A and B) = P(A) * P(B) to test independence. May 06, 2015 Example 3 Are events A and B independent. a) P(A) = 0.8 P(B) = 0.4 P(A and B) = 0.2 These are Not Independent because P(A)* P(B) = (0.8)(0.4) = 0.32 and this is not the same as P(A and B) = 0.2 provided. b) P(A) = 0.6 P(B)=0.5 P(A and B) = 0.3 These are Independent because P(A) * P(B) = (0.6)(0.5) = 0.30 and this is the same as P(A and B) = 0.3 provided Think Pair share How are mutually exclusive events different from independent events? Example 4 Are events A and B independent? Show work. a. b. Example 5 The twoway frequency table shows data for 120 randomly selected patients who have the same doctor. Determine whether a patient who takes vitamins and a patient who exercises regularly are independent events Unit 13.3A.notebook Example 6 A student wants to know if righthanded people are more or less likely to play a musical instrument than lefthanded people. The student collects data from 250 people, as shown in the twoway frequency table. Determine whether being righthanded and playing a musical instrument are independent events. May 06, 2015 Think pair share What are the two different ways you can show two events are independent? Explain.