MIT Probability - The Science of Uncertainty and Data 6.431x Notes

Probability Models:

• A model consists of:
  • A sample space.
  • Probability laws that assign probabilities to elements or groups of elements in the sample space.

Axioms and Useful Results:

We use set notation to work with probability.

• \[P(\Omega)=1\]
• \(P(A\cup B)=P(A)+P(B)\) for disjoint events A and B.
• \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) for not necessarily disjoint A and B.

Its a good idea to convert events into a form where we have all disjoint sets which then allows us to use the addition axiom.

Discrete Uniform Law / Model:

• If a sample space \(\Omega\) has \(n\) elements and all of them are equally likely.
• If an event \(A\) is a subspace of \(\Omega\) and has \(k\) elements.
• The probability of event \(A\) is \(k.\frac{1}{n}\)