How to calculate from left to right result, step by step E (1 – p)?k=1 =Ž (1 – p)?k1- P k=1k=1

Considering the LHS looping the values from 1 to 4 ∑k=1∞(1-p)2k-1 = 11-p∑k=1∞(1-p)2k LHS…

Answered: Ž (1 – p)?k=1 = -P- Ž (1 – p)²k 2k-1 Ë…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

How to calculate from left to right result, step by step

E (1 – p)?k=1 =
Ž (1 – p)?k
1- P k=1
k=1

Expert Solution

Step 1

Considering the LHS looping the values from 1 to 4

$\begin{array}{rcl} \sum_{k = 1}^{\infty} {(1 - p)}^{2 k - 1} & = & \frac{1}{1 - p} \sum_{k = 1}^{\infty} {(1 - p)}^{2 k} \\ L H S & = & \sum_{k = 1}^{4} {(1 - p)}^{2 k - 1} \\ = & {(1 - p)}^{2 (1) - 1} + {(1 - p)}^{2 (2) - 1} + {(1 - p)}^{2 (3) - 1} + {(1 - p)}^{2 (4) - 1} \\ = & (1 - p) + {(1 - p)}^{3} + {(1 - p)}^{5} + {(1 - p)}^{7} \\ = & (1 - p) [1 + {(1 - p)}^{2} + {(1 - p)}^{4} + {(1 - p)}^{6}] \end{array}$