Probability

Probability

1. Odds

   • "The probabilities to win against the probabilities to lose". Odds compare the number of favorable outcomes to the number of unfavorable outcomes. 

2. Basic Concepts

   • Outcomes
     • The possible results of the experiment.
   • Sample Space
     • The set of all possible outcomes. 
   • Event
     • Any such subset of the sample space.
     • involving "not" and "or"
       • Addition rule of probality
         • P(A or B) = P(A) + P(B) - P(A and B)
         • Mutually exclusive events
           • Mutually exclusive events cannot occur simultaneously. P(A or B) = P(A) + P(B)
       • Probability of the complement
         • P(not E) = 1 - P(E)
     • involving "and"
       • Multiplication rule
         • for independent events
           • P(A and B) = P(A) ∙ P(B)
         • for any two events
           • P(A and B) = P(A) ∙ P(B|A)
       • Conditional Probability formula
         • The probability of event B, computed on the assumption that even A has happened. P(B|A) = P(A∩B) / P(A)
     • Independent event
       • P(B|A) =  P(B) or, equivantly P(A|B) = P(A)
   • Experiments
     • Any observation, or measurament of a random phenomenon.

3. Formulas

   • Theoretical
     • P(E) = number of favorable outcomes / total number of outcomes =  n(E) / n(S)
   • Empirical
     • P(E) = number of times event E ocurred / number of times the experiment was performed

4. Set Operations

   • Intersection
     • A ∩ B = { x|x ∈ A and x ∈ B }
   • Complement
     • A' = { x|x ∈ U and x ∉ A }
   • Union
     • A U B = { x|x ∈ A or x ∈ B }
   • Difference
     • A - B = { x|x ∈ A and x ∉ B }

5. Properties

   • certain event
     • P(S) = 1 The probabilty of a certain event is 1.
   • impossible event
     • P(Ø) = 0 The posibility of an impossible event is 0.
   • probability value
     • 0 ≤ P(E) ≤ 1 The propabilty of an event is a number from 0 trough 1, inclusive.