6.3 Probability

6.3.1 Axioms of Probability

  • Axiom I: The probability of an event happening positive, or zero. $P[A]\leq0$
  • Axiom II: The total (added) probability of each event in a set is equal to one. $P[S]=1$
  • Axiom III: If two subsets have no elements in common, then the probability of the two sets combined into one set is the individual probability of the individual sets added together. $\text{If } A\cap{B}= \emptyset\text{ then } P[A\cup{B}] = P[A]+P[B]$

6.3.2 Terms & Definitions

  • Compliment: $S^c$ takes the exact opposite of the set it's acting on. A complimented compliment makes no compliment at all.
  • DeMorgan's Rule: An equivalent statement of set operators can be found by complimenting each set, flipping the operators, and complimenting the final output.
  • Intersection Operator: The intersection operator $\cap$, finds only the elements in common between two sets. This is the same as the logical AND function
  • Sample Space: The sample space, denoted as $S$ is a set of all possible outcomes of an event. It can either be a list, like $S=\{ 1,2,4,8\}$, or a property, or list or properties that defines and fully constrains a variable. An example of the latter is $S=\{ x:x \text{ is even and } 0\leq{x}\leq{10}\}$ or $S=\{ (x,y): x+y=1, 0\leq{x}\leq{1},0\leq{y}\leq{1}\}$
  • Union Operator: The union operator, $\cup$, combines to sets together - but keep in mind it does not repeat elements which are common between the two sets. This is the same as a logical OR function.