Ch1: Experiment, sample space and probability

Experiment and sample space

Sample Space

The set $S$ of all possible outcomes of an experiment.

Event

An event $A$ is a subset of the sample space $S:A\subset S$.

Mutually Exclusive

• A and B are mutually exclusive (or disjoint) events if $A\cap B=\mathrm{\varnothing }$

• The events: $A_1,A_2,...A_n$ are called mutually exclusive (or disjoint) if:

  $$A_i\cap A_j=\mathrm{\varnothing }$$

  for every possible combination of (i, j) for which $i\ne j$

Notation:

• $\{A_i\}$ is a sequence of events.
• The Intersection of the sequence is written as $\bigcap _{i=1}^n{A}_i$
  • If infinite $\bigcap _{i=1}^{\mathrm{\infty }}{A}_i$
  • $\bigcap _i{A}_i$ is the event that occurs if each of the events Ai occurs.
• The Union of these sequence is written as $\bigcup _{i=1}^n{A}_i$
  • If infinite $\bigcup _{i=1}^{\mathrm{\infty }}{A}_i$
  • $\bigcup _i{A}_i$ is the event that occurs if at least one of the events Ai occurs.

Partition

The sequence of events $\{A_i\}$ is a partition of the event B if the events Ai are mutually exclusive and $B=\bigcup _i{A}_i$

Properties of combinations of events

• $A\cap (B\cup C)=(A\cap B)\cup (B\cap C)$
• $A\cup (B\cap C)=(A\cup B)\cap (B\cup C)$
• $A\cup B=A\cup (\bar{A}\cap B)$
• $B=(A\cap B)\cup (\bar{A}\cap B)$
• $\overline{A\cup B}=\bar{A}\cap \bar{B}$
• $\overline{A\cap B}=\bar{A}\cup \bar{B}$

Symmetric Probability Spaces

Probability P of event A

$0\le P(A)\le 1,\mathrm{\forall }A:A\subset S$

$$P(A)=\frac{\text{times A occurred}}{\text{possible outcomes}}$$

Symmetric Probability Space

If S is a finite sample space of an experiment and the probabilities P(A) of events A are defined according to Laplace's definition (outcomes are equally likely) the pair (S, P) is called a symmetric probability space.

The definition of Laplace applies when during an experiment an element is chosen arbitrarily or at random from a finite sample space.

Properties of a symmetric probability space

• $P(A)\ge 0,\mathrm{\forall }A$
• $P(S)=1$
• $A\subset B,then,P(A)\le P(B)$
• $P(\bar{A})=1-P(A)$
• If $A_1,A_2,...A_n$ are mutually exclusive then $P(\bigcap _{i=1}^n{A}_i)=\sum _{n=1}^{n}P(A_i)$

Probabilistic experiments

Definition

An experiment is probabilistic or stochastic if you cannot know the outcome of the experiment ahead of time. E.g.: A diceroll or a toss of a coin.

Relative frequency and the empirical law of large numbers

Definition

Assume that we have an experiment with sample space S which we can repeat arbitrarily often. If the event A occurred n(A) times in total with n repetitions, then we can define:

$$f_n(A)=\frac{n(A)}{n}$$

As the relative frequency (or frequency quotient) of A in n repetitions.

Kolmogorov's Axioms

Definition

Consider an experiment with a random non-empty sample space S. A function P which assigns a real number P(A) to every event $A\subset S$, is called a probability or probability measure on S if:

1. $P(A)\ge 0$ for every event A
2. $P(S)=1$
3. For every countable sequence of mutually exclusive events $A_1,A_2,...,A_{n}P(\bigcap _i{A}_i)=\sum _{i}P(A_i)$

Probability Space

When S is a sample space and P is the probability on S then we call the pair (S, P) a probability space.

Properties

• $P(\mathrm{\varnothing })=0$
• $P(\bar{A})=1-P(A)$
• For two events A and B with $A\subset B$ we have $P(A)\le P(B)$
• For two events A and B (not necessarily mutually exclusive): $P(A\cup B)=P(A)+P(B)-P(A\cap B)$