| We are dealing here with mass phenomena and their “average” behavior. The physical system that creates the OBSERVABLE phenomena is usually assumed to be CONSISTENT and the observation {either over time (a sequence) or among a simultaneous population (a SET, an ENSEMBLE)} will have statistics that are also consistent. We need a theory that describes and predicts these statistics. |
| Physical | ||
| Experiment (frequency ratio, apostiori) | ||
| Classical Definition (# of trials large) | ||
| Conceptual - Axiomatic | ||
| Axioms and Reasoning (heavy math) | ||
| Prediction | ||
| Symmetry (thought experiment, apriori) | ||
| The Sample Space | ||
| S is the collection or set of all possible outcomes of an experiment. It is the universal set for this experiment. | ||
| Discrete Sample Space: Finite or countably infinite set. (e.g. faces of a die, the positive integers, students in this class) | ||
| Continuous Sample Space: not countable. (e.g. the real line, battery voltage in a car) | ||
| Definition | ||
| To each outcome in S we associate a non-negative number Pn = P(xn) such that : | ||
| 0<Pn<1 and Sum[ P(xn)] = 1 | ||
| xn is in S | ||
| Sum is over all elements of S | ||
| {xn} is collectively exhaustive and mutually exclusive | ||
| P(A) is greater than or equal to 0 | |
| P(S) = 1 | |
| P(A + B) = P(A) + P(B) - P(A*B) | |
| P({}) = 0 | |
| Mutually Exclusive Events: P(A + B) = P(A) + P(B) |
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| The three axioms of probability |