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