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
Outcome
A particular result of an experiment. ␣ Ex: rolling a 3 on a dice is an outcome
Event
A collection of one or more outcomes of an experiment. Ex: 1,4,5
Sample space
The collection of all outcomes of an experiment␣ Denoted by S Ex:Is sample space an event? Yes.
Probability
- A number between 0 and 1
- Reflects the likelihood of occurrence of an event
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
Empirical Probability
Definition: Number of times an event occurred divided by thetotal number of trials
Properties: determined a posteriori
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=
Mutually Exclusive Events
Events with no common outcomes
Occurrence of one event excludes occurrence of theother
Union of Events
The union of two events, A & B, contains all the outcomes in both events.
Denoted as A∪B
Examples:
A = {1, 2, 4, 7, 9}, B = {2, 3, 4, 5, 6}
A∪B= 1,2,3,4,5,6,7,9
A
A = {Blue, Purple, Red, White}, B = {Green, Red, Yellow}
A∪B= Red
Partition of Sample Space
Events are both mutually exclusive and collectively exhaustive
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.
Addition Law
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Addition Law; Mutually Exclusive Events Case
If A and B are mutually exclusive, then P(A ∪ B) = P(A) + P(B)
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
Probability Matrix
Conditional Probability
P(A| B) = P(A∩B) / P(B)
Multiplication Law
P(A∩ B) = P(A)⋅P(B | A)
Multiplication Law; Independent Events Case
If A and B are independent, then
P(A ∩ B) = P(A)· P(B)
Random Variables (R.V)
a variable whose value may change from one experimental unit to another
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.
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.
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
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