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 $E ( X ) = \sum _ { n = 1 } ^ { \infty } n \frac { 3 { n } } { 4 ^ { n + 1 } } = \frac { 1 } { 4 } \sum _ { n = 1 } ^ { \infty } n \left( \frac { 3 } { 4 } \right) ^ { n } = 3$ correct?

Question

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 $E ( X ) = \sum _ { n = 1 } ^ { \infty } n \frac { 3  { n } } { 4 ^ { n + 1 } } = \frac { 1 } { 4 } \sum _ { n = 1 } ^ { \infty } n \left( \frac { 3 } { 4 } \right) ^ { n } = 3$ correct?

Quizplus · Accepted Answer

The given expression correctly calculates the expected number of consecutive red balls between occurrences of blue balls using the geometric series formula.

Suppose That a Bag Holds 3 Blue Balls and One