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Probabilistic spaceΒΆ

Probability Space: Theory and ExamplesΒΆ

A probability space is a mathematical triplet \((\Omega, \mathcal{F}, P)\) that fully describes a random process. It consists of three fundamental components.

1. Theoretical DefinitionΒΆ

The Sample Space (\(\Omega\))ΒΆ

The set of all possible distinct outcomes of the experiment.

The Sigma-algebra (\(\mathcal{F}\))ΒΆ

A collection of subsets of \(\Omega\), known as events. This collection defines "what can be measured." It must satisfy three properties:

  • Non-empty: \(\Omega \in \mathcal{F}\).
  • Closed under complement: If event \(A\) is in \(\mathcal{F}\), then "not A" (\(A^c\)) is also in \(\mathcal{F}\).
  • Closed under countable unions: If you have a sequence of events, their union is also an event.

The Probability Measure (\(P\))ΒΆ

A function \(P: \mathcal{F} \to [0, 1]\) that assigns a probability to every event in \(\mathcal{F}\). It must satisfy:

  • \(P(\Omega) = 1\).
  • \(P(\emptyset) = 0\).
  • Countable Additivity: For mutually exclusive events \(A_1, A_2, \dots\):
\[ P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i) \]

2. Example: Single Coin TossΒΆ

This is the simplest non-trivial example.

  • Experiment: Tossing a fair coin once.
  • Sample Space (\(\Omega\)): There are two possible outcomes: Heads (\(H\)) and Tails (\(T\)).
\[ \Omega = \{H, T\} \]
  • Sigma-algebra (\(\mathcal{F}\)): We need to list all possible events. Since the space is finite and small, we take the Power Set (the set of all subsets).
\[ \mathcal{F} = \big\{ \emptyset, \{H\}, \{T\}, \{H, T\} \big\} \]

Interpretation of \(\mathcal{F}\) elements:

  • \(\emptyset\): The impossible event (neither Heads nor Tails occurs).
  • \(\{H\}\): The event "Heads occurs".
  • \(\{T\}\): The event "Tails occurs".

  • \(\{H, T\}\): The certain event (either Heads or Tails occurs). This is equivalent to \(\Omega\).

  • Probability Measure (\(P\)): Assuming a fair coin:

\[ P(\{H\}) = 0.5, \quad P(\{T\}) = 0.5 \]
\[ P(\emptyset) = 0, \quad P(\Omega) = 1 \]

3. Example: Rolling a DieΒΆ

A slightly more complex discrete example.

  • Experiment: Rolling a standard six-sided die.
  • Sample Space (\(\Omega\)):
\[ \Omega = \{1, 2, 3, 4, 5, 6\} \]
  • Sigma-algebra (\(\mathcal{F}\)): Usually, for discrete spaces, we use the Power Set (\(2^\Omega\)), which contains all possible combinations of outcomes. Size of \(\mathcal{F} = 2^6 = 64\) events.

Examples of specific events in \(\mathcal{F}\):

  • Elementary events: \(\{1\}, \{2\}, \dots\) (Rolling a specific number).

  • Compound events:

    • \(A = \{2, 4, 6\}\): The event "Rolling an even number".
    • \(B = \{5, 6\}\): The event "Rolling more than 4".

  • Probability Measure (\(P\)): For a fair die, we assign a probability of \(\frac{1}{6}\) to each elementary outcome. The probability of any event \(E\) is the sum of the probabilities of the outcomes in \(E\).

Calculations:

\[ P(\{1\}) = \frac{1}{6} \]
\[ P(\text{Even}) = P(\{2, 4, 6\}) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = 0.5 \]

Probability Simulator

1. Coin Toss (Bernoulli Trial)

Theoretical Probability: P(Heads) = 0.5, P(Tails) = 0.5

Result: -
Heads:
0 (0%)
Tails:
0 (0%)

Total Tosses: 0

2. Die Roll (Uniform Distribution)

Theoretical Probability: P(k) β‰ˆ 16.67% for k ∈ {1..6}

Result: -

Total Rolls: 0