Quantitation Probabilities

For looking at probability of events, we set the following defintions. Let $A$ be an event and $I$ the given information:

• $\mathbb{P}(A | I)$: The probability that $A$ is true given the information $I$.
• $\mathbb{P}(AB | I)$: The probability that $A$ and $B$ are true given the information $I$.
• $\mathbb{P}(A |B I)$: The probability that $A$ is true given the information $I$ and that $B$ is true.

Boolean Operations:

Not $A$ : $\bar{A}$	1	0
$A$	0	1

$A$ and $B$ : $AB$	0	0	0	1
$A$	0	0	1	1
$B$	0	1	0	1

$A$ or $B$ : $A + B$	0	1	1	1
$A$	0	0	1	1
$B$	0	1	0	1

$A$ implies $B$ : $A \rightarrow B$	1	1	0	1
$A$	0	0	1	1
$B$	0	1	0	1

The usage of such tables for larger systems is called the disjunctive normal form, where AND’s and OR’s are connected to make new statements.

Rules of probabilities

1. Probabilites are real numbers between 0 (false) and 1 (true)
2. For false/true statements, the rules reduce to the rules of Boolean logic.
3. Consistency: If a probability can be derived in different was, it should give the same results.

The two main rules of probability, wherefrom every other rule can be derived are:

1. \[\mathbb{P}(A | I) + \mathbb{P}(\bar{A} | I) = 1\]
2. \[\mathbb{P}(AB | I) = \mathbb{P}(A | BI)\mathbb{P}(B | I) = \mathbb{P}(B | AI)\mathbb{P}(A | I)\]

For example we have

\[\mathbb{P}(A + B | I) = \mathbb{P}(A | I) + \mathbb{P}(B | I) - \mathbb{P}(AB | I)\]

Proof

$$ \begin{align*} \mathbb{P}(A + B | I) &=\mathbb{P}(\overline{\bar{A}\bar{B}} | I) \\ &=1 - \mathbb{P}(\bar{A}\bar{B} | I) \\ &= 1 - (\mathbb{P}(\bar{A} | \bar{B}I)\mathbb{P}(\bar{B} | I)) \\ &= 1 - (1 - \mathbb{P}(A | \bar{B}I))\mathbb{P}(\bar{B} | I)) \\ &= 1 - \mathbb{P}(\bar{B} | I) + \mathbb{P}(A | \bar{B}I)\mathbb{P}(\bar{B} | I) \\ &= \mathbb{P}(B | I) + (\mathbb{P}(\bar{B} | AI) \mathbb{P}(A |I)) \\ &= \mathbb{P}(B | I) + (1 - \mathbb{P}(B | AI)) \mathbb{P}(A |I) \\ &= \mathbb{P}(B | I) + \mathbb{P}(A |I) - \mathbb{P}(B | AI)\mathbb{P}(A |I) \\ &= \mathbb{P}(B | I) + \mathbb{P}(A |I) - \mathbb{P}(AB | I) \quad q.e.d \end{align*} $$

Now lets imagine we have $n$ possible outcomes, $A_1, …, A_n$, where each event is mutually exclusive, meaning that if $A_i$ is true, then all others are false and that they are exhaustive, meaning that one out of all events must be true. With this we get the following properties:

\[\mathbb{P}(A_i A_j | I) = 0 \ \forall i \neq j \quad \text{and} \quad \sum_{i = 1}^n \mathbb{P}(A_i | I) = 1\]

Because we provide no further information with which we can distinguish event $A_i$ from $A_j$, we treat them all as the same and we finally get:

\[\mathbb{P}(A_i | I) = \frac{1}{n} \ \forall i\]