Probability

What Is Probability?

Probability quantifies the chance that an event will occur. It represents uncertainty and is used when exact prediction is impossible. Probabilities are numbers between 0 and 1

Discrete Outcomes

Occur when the number of possible outcomes is finite or countable.
Example: Number of heads in coin tosses.
Example: Number of people on telephone calls in a given period.

Continuous Outcomes

Outcomes can take infinitely many values within a range.
Example: Length of time a phone call lasts (between 0 and 10 minutes).
Cannot assign nonzero probability to a single exact value in continuous cases.

Probabilistic Modeling

Building a probability model means mapping a real-world situation to a mathematical structure (e.g., random variables, distributions). Models often make assumptions — e.g., independence (one event doesn’t affect another).

Illustration: Probability of k heads in N coin tosses:
$P[k] = \binom{N}{k}p^k(1-p)^{N-k}$

Analysis vs. Computer Simulation

Traditional analysis uses mathematical formulas to derive results. Computer simulation uses repeated computational experiments to approximate probabilities and build intuition.

Role of Simulation

Helpful when analytical solutions are difficult or intractable.
Visual or numerical results from simulation help validate or illustrate analytical theory.

Key Takeaways

Concept	Brief Summary
Probability	A measure of how likely an event is; values between 0 and 1
Discrete vs. Continuous	Discrete: countable outcomes; Continuous: uncountably infinite outcomes.
Modeling	Abstracting real phenomena into probabilistic terms.
Simulation	Complementary to analysis; enhances intuition.

References

Introduction of Intuitive Probability and Random Processes Using MATLAB by Steven M. Kay (Springer, 2006)