Prerequisites & Notation

Before You Begin

This chapter applies the probability axioms from Chapter 1 and the conditional probability machinery from Chapter 2 to engineering systems. The chapter is entirely self-contained once you are comfortable with the items below.

Kolmogorov axioms, inclusion-exclusion principle (FSP Ch. 1)(Review fsp-ch01)
Self-check: Can you state the inclusion-exclusion formula for three events and identify which terms have which sign?
Conditional probability and independence (FSP Ch. 2)(Review fsp-ch02)
Self-check: Can you argue from first principles why $P(A \cap B) = P(A)\,P(B)$ when $A$ and $B$ are independent?
Geometric series and harmonic numbers
Self-check: Do you know that $H_n = \sum_{k=1}^n 1/k \approx \ln n$ for large $n$ ?
Basic combinatorics: permutations, combinations (FSP Ch. 1.4)(Review fsp-ch01)
Self-check: Can you quickly compute $\binom{n}{k}$ and explain the difference between sampling with and without replacement?

Notation for This Chapter

Symbols introduced or prominently used in this chapter. See also the NGlobal Notation Table master table.

Symbol	Meaning	Introduced
$R_s$	Reliability of a system (probability of correct operation)	s01
$R_i$	Reliability of component $i$ (each component fails independently)	s01
$A_i$	Event that component $i$ is working	s01
$F_i$	Event that component $i$ has failed ( $F_i = A_i^c$ )	s01
$H_n$	Harmonic number $H_n = \sum_{k=1}^n 1/k$	s03
$T_n$	Coupon collector stopping time: first time all $n$ coupons seen	s03
$C$	Channel capacity (bits per channel use)	s02
$R$	Transmission rate (bits per channel use)	s02

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