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a. Differentiate the series 1 + x + x² + ·.. + x" +.. х to obtain a series for 1/(1 – x)². b. In one throw of two dice, the probability of getting a roll of 7 is p = 1/6. If you throw the dice repeatedly, the probability that a 7 will appear for the first time at the nth throw is q"-'p, where q = 1 – p = 5/6. The expected number of throws un- til a 7 first appears is Enq"-'p. Find the sum of this series. c. As an engineer applying statistical control to an industrial operation, you inspect items taken at random from the as- sembly line. You classify each sampled item as either “good" or “bad." If the probability of an item’'s being good is p and of an item's being bad is q = 1 – p, the probability that the first bad item found is the nth one inspected is p"-'q. The average number inspected up to and including the first bad item found is E-¡np"-'q. Evaluate this sum, assuming 0

a. Differentiate the series 1 + x + x² + ·.. + x" +.. х to obtain a series for 1/(1 – x)². b. In one throw of two dice, the probability of getting a roll of 7 is p = 1/6. If you throw the dice repeatedly, the probability that a 7 will appear for the first time at the nth throw is q"-'p, where q = 1 – p = 5/6. The expected number of throws un- til a 7 first appears is Enq"-'p. Find the sum of this series. c. As an engineer applying statistical control to an industrial operation, you inspect items taken at random from the as- sembly line. You classify each sampled item as either “good" or “bad." If the probability of an item’'s being good is p and of an item's being bad is q = 1 – p, the probability that the first bad item found is the nth one inspected is p"-'q. The average number inspected up to and including the first bad item found is E-¡np"-'q. Evaluate this sum, assuming 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
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a. Differentiate the series 1 + x + x² + ·.. + x" +.. х to obtain a series for 1/(1 – x)². b. In one throw of two dice, the probability of getting a roll of 7 is p = 1/6. If you throw the dice repeatedly, the probability that a 7 will appear for the first time at the nth throw is q"-'p, where q = 1 – p = 5/6. The expected number of throws un- til a 7 first appears is Enq"-'p. Find the sum of this series. c. As an engineer applying statistical control to an industrial operation, you inspect items taken at random from the as- sembly line. You classify each sampled item as either “good" or “bad." If the probability of an item’'s being good is p and of an item's being bad is q = 1 – p, the probability that the first bad item found is the nth one inspected is p"-'q. The average number inspected up to and including the first bad item found is E-¡np"-'q. Evaluate this sum, assuming 0 <p < 1.
Transcribed Image Text:a. Differentiate the series 1 + x + x² + ·.. + x" +.. х to obtain a series for 1/(1 – x)². b. In one throw of two dice, the probability of getting a roll of 7 is p = 1/6. If you throw the dice repeatedly, the probability that a 7 will appear for the first time at the nth throw is q"-'p, where q = 1 – p = 5/6. The expected number of throws un- til a 7 first appears is Enq"-'p. Find the sum of this series. c. As an engineer applying statistical control to an industrial operation, you inspect items taken at random from the as- sembly line. You classify each sampled item as either “good" or “bad." If the probability of an item’'s being good is p and of an item's being bad is q = 1 – p, the probability that the first bad item found is the nth one inspected is p"-'q. The average number inspected up to and including the first bad item found is E-¡np"-'q. Evaluate this sum, assuming 0 <p < 1.
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