Probability theory is a branch of mathematics concerned with the science of chance. Probability describes the real world. The data we collect and models we build involve statistics and statistical learning.

Basic terminology

A random experiment is one where there’s a varied outcome each time. We define the sample space as the set of all possible outcomes. We define the likelihood of each outcome in trials as , and the relative frequency of an event in trials as , such that the probability of an event is:

In practice, an infinite number of trials is impossible, so we instead define several axioms to model probability:

  • The probability of an event , .
  • Total probability, i.e., of the sample space, is .
  • If events are mutually exclusive, then .

We have some handy formulas, but it’s nice to first define some terms.

  • An event is a condition on an outcome. For example, an event such that a roulette result is even and is non-zero.
    • If an event produces one outcome (instead of a subset of ), then we call this an elementary event.
    • Complicated events are a set union of elementary events.
  • The complement of an event is the event that does not occur. Complementary events are mutually exclusive and exhaustive events; i.e., their probabilities will together sum to 1.
    • We additionally define the complement of as or or . Then, .
  • Mutual exclusivity of two events suggest that they cannot occur at the same time.
  • Events are exhaustive if at least one event must occur, i.e., when rolling a six-sided die.
  • Events are independent if one event occurring does not affect the probability of another event occurring.
  • Conditional probability is the probability of event occurring given that event has occurred.

And some useful axioms:

  • For complementary events, .
  • For combined events, .
  • For mutually exclusive events, . In fact, if is the union of distinct mutually disjoint subsets, then is the sum of the probability of each subset.
  • For conditional events, .
  • For independent events, .

Extensions

It also helps us to define a few formulas given certain situations. If every outcome is equally likely, then the probability of an event is:

Constructing statements for the probability of certain events often takes combinatorial approaches, including permutations and combinations.

For situations that involves us successively making decisions (like sampling from a set of items more than once). In this case, we can refer to the rule of products. This holds regardless of how the preceding steps were formed:

If an operation consists of steps, and the first step can be performed in ways, second in ways, th step in ways, then the entire operation is performed in: .

From here, we can obviously tell that if , then the operation has possibilities. If we don’t replace an element in the sample space (i.e., we can’t repeat an element), then we have , and the operation has possibilities.

Sub-pages

Resources

  • Probability, Statistics, and Random Processes for Electrical Engineering, by Alberto Leon-Garcia
  • Probability Theory: the Logic of Science, by E.T. Jaynes

See also