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  1. Experiment
    A process that leads to the occurrenceof one and only one of several possible observations.

    Ex: Rolling a die and recording the number of spots on the top face of the die
  2. Outcome
    A particular result of an experiment. ␣ Ex: rolling a 3 on a dice is an outcome
  3. Event
    A collection of one or more outcomes of an experiment. Ex: 1,4,5
  4. Sample space
    The collection of all outcomes of an experiment␣ Denoted by S Ex:Is sample space an event? Yes.
  5. Probability
    • - A number between 0 and 1
    • - Reflects the likelihood of occurrence of an event
  6. Classical Probability
    • Assumption:
    • Definition: Number of outcomes leading to the event divided by the total number of possible outcomes

    • Examples: probability of a head on a coin flip
    • probability of a 3 on a die roll =
    • Probability of S, P(S) =
    • Properties: determined a priori
    • objective
  7. Empirical Probability
    Definition: Number of times an event occurred divided by thetotal number of trials

    Properties: determined a posteriori
  8. Intersection of Events
    The intersection of two events, A & B, contains only the common outcomes in both events.

    Denoted as A∩B

    Examples: A = {1, 2, 4, 7, 9}, B = {2, 3, 4, 5, 6} A∩B=␣ A = {Blue, Purple, Red, White}, B = {Green, Red, Yellow} A∩B=
  9. Mutually Exclusive Events
    • Events with no common outcomes
    • Occurrence of one event excludes occurrence of theother
  10. Union of Events

    The union of two events, A & B, contains all the outcomes in both events.

    Denoted as A∪B
    A = {1, 2, 4, 7, 9}, B = {2, 3, 4, 5, 6}

    A∪B= 1,2,3,4,5,6,7,9

    A = {Blue, Purple, Red, White}, B = {Green, Red, Yellow}

    A∪B= Red
  11. Partition of Sample Space
    Events are both mutually exclusive and collectively exhaustive
  12. Complementary Events
    The complement of an event A is the event that A does not occur

    Denoted by Ac P(A) + P(Ac) = 1

    A and Ac form a partition.
  13. Addition Law
    • P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
    • Image Upload 1
  14. Addition Law; Mutually Exclusive Events Case
    • If A and B are mutually exclusive, then P(A ∪ B) = P(A) + P(B)
    • Image Upload 2
  15. Independent Events
    Occurrence of one event does not affect the occurrence/nonoccurrence of the other event

    • The conditional probability of A given B is equal to the marginal probability of A.
    • P(A | B)=P(A)

    Likewise, P(B | A)=P(B)

    • The pizza represents the sample space
    • You are blindfolded and asked to select a slice of
    • pizza
    • Event A = your slice contains pepperoni
    • Event B = your slice contains mushroom
  16. Probability Matrix
    Image Upload 3
  17. Conditional Probability
    P(A| B) = P(A∩B) / P(B)
  18. Multiplication Law
    P(A∩ B) = P(A)⋅P(B | A)
  19. Multiplication Law; Independent Events Case
    If A and B are independent, then
    P(A ∩ B) = P(A)· P(B)
  20. Random Variables (R.V)
    a variable whose value may change from one experimental unit to another
  21. Discrete Distributions
    • Discrete r.v. — takes on a finite or countably infinite number of possible values
    • - Number of bad checks received by a restaurant
    • - Number of absent employees on a given day␣

    Discrete distribution — probability distribution of a discrete r.v.
  22. Continuous Distributions
    • Continuous r.v. — takes on values at every point over a given interval
    • - Elapsed time between arrivals of bank customers
    • - Percentage of the labor force that is unemployed

    Continuous distribution — probability distribution of a continuous r.v.
  23. Binomial Distribution
    • Binomial Distribution
    • - Model experiment involving n independent identical trials
    • - Each trial results in either success (S) or failure (F)

    • Examples
    • ␣ Flipping a fair coin with “success = heads”
    • ␣ Rolling a die with “success = six ”
    • ␣ Conducting political opinion poll with “success = voting for a specific candidate”
    • ␣ Define X as total # of successes in n trials
  24. Normal Distribution
    • Widely used to model natural characteristics ␣ height, weight, length ␣
    • IQ scores ␣ years of life expectancy
    • ␣ Many variables in business and industry are also normally distributed. ␣ annual cost of household insurance ␣ cost per square foot of renting warehouse space
    • ␣ amount of fill in soda cans ␣ Also referred to as Gaussian distribution
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