Probability

Engineering Statistics - STAT3115 - 1 February 2018

Subjective Interpretation

  • P(Impossibility) = 0
  • P(Win the Lottery) ≈ 0
  • P(Sure Thing) = 1
  • P(Death and Taxes) ≈ 1 (basically assured)
  • P(“Heads on a coin toss”) = 0.5
  • P(Gender of newborn is Female) ≈ 0.5

Overcounting

  • It is when you are doing unions and you count the same data more than once
  • Happens when two data sets have overlap
  • To counter for this, with two data sets
    • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
  • To counter for this, with three data sets
    • P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)

Laws/ Identities

Commutative Laws

  • A ∪ B = B ∪ A
  • A ∩ B = B ∩ A

Associative Laws

  • A ∪ B ∪ C = (A ∪ B) ∪ C = A ∪ (B ∪ C)
  • A ∩ B ∩ C = (A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributive Laws

  • (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)
  • (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)

DeMorgan’s Laws

  • (A ∩ B)’ = A’ ∪ B’
  • (A ∪ B)’ = A’ ∩ B’

Find the probability

  1. Define the experiment
  2. List the simple events associated with the experiment and test each to make certain that it cannot be decomposed; this defines the sample space S
  3. Assign reasonable Probabilities to the outcomes in S
  4. Define the event of interest, A, as a union of simple events or a collection of outcomes
  5. Fine P(A) by summing the probabilities of the simple events whose outcomes are in A