Quantum Computing and Communications: An Engineering Approach

12

Mathematical Background

Proofs that odd numbers are prime:

• Mathematician: 1 is prime, 3 is prime, 5 is prime, 7 is prime, therefore, by induction, all odd numbers are prime.
• Physicist: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is a bad data point, 11 is prime, 13 is prime…
• Engineer: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is approximately prime, 11 is prime, 13 is prime…
• Computer scientist:¹ 1 is prime, 1 is prime, 1 is prime, 1 is prime…

12.1 BASIC PROBABILITY THEORY

12.1.1 Characterization of random events

Let A and B denote two random events. If the corresponding probabilities are represented by P(A) and P(B) then we have several important relations:

• OR connection: P (A ∨ B) = P(A) + P(B) − P(A ∧ B). If P(A ∧ B) = 0 then P(A ∨ B) = P(A) + P(B).
• Conditional probability: .
• Independent events: If A and B are independent if and only if P(A ∧ B) = P(A) · P(B) then P(A|B) = P(A).
• Law of total probability: .
• Bayes formula: .

A group of mutually excluding probabilistic events {a} belonging to the same observable is represented by means of random variables in probability theory. If variable A stands for a certain random event then the probability of obtaining A = ...