# 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* = ...

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