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Suppose That a Bag Holds 3 Blue Balls and One

Question 17

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Suppose that a bag holds 3 blue balls and one red ball. We pull a ball from the bag at random, return it and then repeat the process. Suppose we continue pulling balls until the blue ball is drawn and then we observe the number of consecutive red balls drawn. What is the average number of red balls between occurrences of blue balls? Is E(X)=∑n=1∞n3n4n+1=14∑n=1∞n(34)n=3E ( X ) = \sum _ { n = 1 } ^ { \infty } n \frac { 3 { n } } { 4 ^ { n + 1 } } = \frac { 1 } { 4 } \sum _ { n = 1 } ^ { \infty } n \left( \frac { 3 } { 4 } \right) ^ { n } = 3 correct?

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