Probability

Understanding Probability

Probability quantifies uncertainty: given a set of possible outcomes, it assigns each a number between 0 and 1 representing how likely it is. Discrete probability — where outcomes can be counted rather than measured continuously — connects directly to the combinatorics covered in the Counting module, since most probabilities here reduce to counting favorable outcomes over total outcomes.

Key Definitions & Formulas

• Sample space: the set of all possible outcomes of an experiment.
• Event: any subset of the sample space.
• P(A) = favorable outcomes / total outcomes, for equally likely outcomes.
• Independence: events A and B are independent if P(A ∩ B) = P(A)·P(B).
• Conditional probability P(A|B): the probability of A given that B has already occurred, = P(A ∩ B) / P(B).
• Bayes' theorem: relates P(A|B) to P(B|A), letting you "flip" conditional probabilities.

Worked Example

Rolling two fair dice, what's the probability the sum is 7? There are 36 equally likely outcomes total, and exactly 6 of them sum to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1). So P(sum=7) = 6/36 = 1/6 — the most likely single sum when rolling two dice.

Where This Is Used

• Risk assessment in insurance and finance.
• Machine learning models (naive Bayes classifiers, probabilistic reasoning).
• Genetics and inheritance probability calculations.
• Game design and gambling odds calculation.