Explanation Definition: If X is a random variable for the number of successes in n independent and identical repetitions of an experiment with success and failure as two possible outcomes and if p is a probability of success, then the probability for k success is defined as, P ( X = k ) = ( n k ) p k ( 1 − p ) n − k Here, k = 0 , 1 , ⋅ ⋅ ⋅ , n . Calculation: Consider the widgets of the Ampex Widget Company in which eighty percent of the widgets meet the specifications of the customers. If six widgets are tested, then the probability that three or more of the widgets fails to meet the specifications is required. This is a binomial distribution because there are fixed number of trials. The trails are independent and the probability is the same for all trials. Define X as the number of widgets which fails to meet the specifications. As the probability of meeting the specifications is 0.80 , the probability of not meeting the specifications will be 1 − 0.80 = 0.20 . So, n = 6 , X = 3 , p = 0.2 . Use the probability formula and substitute these values to calculate the required probability. P ( X ≥ 3 ) = P ( X = 3 ) + P ( X = 4 ) + P ( X = 5 ) + P ( X = 6 ) = ( 6 3 ) ( 0.2 ) 3 ( 1 − 0.2 ) 6 − 3 + ( 6 4 ) ( 0.2 ) 4 ( 1 − 0.2 ) 6 − 4 + ( 6 5 ) ( 0.2 ) 5 ( 1 − 0.2 ) 6 − 5 + ( 6 6 ) ( 0.2 ) 6 ( 1 − 0.2 ) 6 − 6 Use the combination formula ( n r ) = n ! r ! ( n − r ) ! and simplify

Nature of Mathematics (MindTap Course List)

13th Edition

ISBN: 9781133947257

Author: karl J. smith

Publisher: Cengage Learning

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Chapter 13.6, Problem 52PS

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The probability that three or more widgets would fail to meet the specifications from a sample of six widgets tested with eighty percent of the widgets of Ampex Widget Company meeting the specifications of its customers.

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Chapter 13.6, Problem 51PS

Chapter 13.6, Problem 53PS

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Nature of Mathematics (MindTap Course List)

Ch. 13.1 - IN YOUR OWN WORDS What is the difference between...Ch. 13.1 - IN YOUR OWN WORDS Define probabilities?Ch. 13.1 - Prob. 3PS Ch. 13.1 - Prob. 4PS Ch. 13.1 - Prob. 5PS Ch. 13.1 - Prob. 6PS Ch. 13.1 - Prob. 7PS Ch. 13.1 - Prob. 8PS Ch. 13.1 - Give the probabilities in Problems 9-11 in decimal...Ch. 13.1 - Give the probabilities in Problems 9-11 in decimal...

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The probability that three or more widgets would fail to meet the specifications from a sample of six widgets tested with eighty percent of the widgets of Ampex Widget Company meeting the specifications of its customers.

Solution Summary: The author calculates the probability that three or more widgets would fail to meet the specifications from a sample of six Widgets tested.

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To find: The probability that three or more widgets would fail to meet the specifications from a sample of six widgets tested with eighty percent of the widgets of Ampex Widget Company meeting the specifications of its customers.

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Chapter 13 Solutions

To find:

The probability that three or more widgets would fail to meet the specifications from a sample of six widgets tested with eighty percent of the widgets of Ampex Widget Company meeting the specifications of its customers.