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Understanding Independent Events

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moriah
16 diapositives · 365 Views · Sep 30, 2014

Understanding Independent Events. Adapted from Walch Education. 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,. Concepts, continued.

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  1. Understanding Independent Events Adapted from Walch Education

  2. 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, 7.1.3: Understanding Independent Events

  3. 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 7.1.3: Understanding Independent Events

  4. 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: 7.1.3: Understanding Independent Events

  5. Note: • 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. 7.1.3: Understanding Independent Events

  6. Probability and relative frequency are related as follows: • 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 7.1.3: Understanding Independent Events

  7. The Addition Rule • 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 7.1.3: Understanding Independent Events

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

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

  10. Step 2 Use the sample space to determine the relevant probabilities. 7.1.3: Understanding Independent Events

  11. Step 2, continued… 7.1.3: Understanding Independent Events

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

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

  14. Try this problem: • 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 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 Landen’s club sandwich on country white bread. • A random customer orders the roasted chicken sandwich on whole wheat bread. 7.1.3: Understanding Independent Events

  16. Thanks for Watching! Ms. Dambreville