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UNDERSTANDING INDEPENDENT EVENTS Adapted from Walch Education

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Key Concepts 7.1.3: Understanding Independent Events 2 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,

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Concepts, continued. 7.1.3: Understanding Independent Events 3 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

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Concepts, continued. 7.1.3: Understanding Independent Events 4 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:

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Note: 7.1.3: Understanding Independent Events 5 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.

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Probability and relative frequency are related as follows: 7.1.3: Understanding Independent Events 6 The probability of an event can be used to predict its relative frequency if the experiment is performed a large number of times 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

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The Addition Rule 7.1.3: Understanding Independent Events 7 Remember: 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

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Practice 7.1.3: Understanding Independent Events 8 Trevor tosses a coin 3 times. Consider the following events. For each of the following pairs of events, determine if the events are independent.

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Step 1 7.1.3: Understanding Independent Events 9 List the sample space. Sample space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

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Step 2 7.1.3: Understanding Independent Events 10 Use the sample space to determine the relevant probabilities. There are 4 outcomes with heads first. There are 4 outcomes with heads second. There are 2 outcomes with exactly 2 consecutive heads.

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Step 2, continued 7.1.3: Understanding Independent Events 11 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.

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Step 3 7.1.3: Understanding Independent Events 12 Use the definition of independence to determine if the events are independent in each specified pair. A and B are independent.

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Conclusion 7.1.3: Understanding Independent Events 13 A and C are independent. B and C are dependent.

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Try this problem: 7.1.3: Understanding Independent Events 14 Landen owns a delicatessen. He collected data on sales of his most popular sandwiches for one week and recorded it in the table below. continued Bread choice Sandwich choice Landens club Turkey melt Roasted chicken Veggie delight Country white4425 8 Whole wheat24282634 Sourdough24272431

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7.1.3: Understanding Independent Events 15 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 Landens club sandwich on country white bread. A random customer orders the roasted chicken sandwich on whole wheat bread.

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Ms. Dambreville Thanks for Watching!