Final answer:
Probability is defined as the likelihood of an event, assigned a number between zero and one. It can be approached classically, where outcomes are equally likely, or subjectively, based on personal belief. Key concepts include composite events, continuous versus discrete sample spaces, and uncountably-infinite sample spaces.
Step-by-step explanation:
The modern definition of probability, within the domain of mathematics, refers to the likelihood of an event occurring, quantified as a number between zero and one, inclusive. There are two primary approaches to defining probability: the classical approach and the subjective approach. The classical approach is objective and quantifiable, based on the assumption that all outcomes in a sample space are equally likely; it's used in situations with a clear-cut set of finite outcomes, like flipping a coin. On the other hand, the subjective approach considers probability as a degree of belief, influenced by an individual's opinion or information available; it's more flexible and applicable when dealing with personal judgements or a lack of symmetry in outcomes.
When discussing types of events and sample spaces within probability:
- A composite event refers to an event composed of multiple simple events.
- An uncountably-infinite sample space would include outcomes that cannot be listed one-by-one (such as all points on a segment of a line), while a discrete sample space consists of countable outcomes.
- A continuous sample space has an infinite number of outcomes, typically represented by measurements rather than counts. This is contrasted with a discrete sample space where outcomes are countable and individual probabilities for each outcome can be assigned.