1. STA 291 Spring 2008
Lecture 7
Dustin Lueker

2. Probability Terminology
Experiment
Any activity from which an outcome, measurement, or other such result is obtained
Random (or Chance) Experiment
An experiment with the property that the outcome cannot be predicted with certainty
Outcome
Any possible result of an experiment
Sample Space
Collection of all possible outcomes of an experiment
Event
A specific collection of outcomes
Simple Event
An event consisting of exactly one outcome
STA 291 Spring 2008 Lecture 7

3. Basic Concepts Let A and B denote two events Complement of A
All the outcomes in the sample space S that do not belong to the even A
P(Ac)=1-P(A)
Union of A and B
A ∪ B
All the outcomes in S that belong to at least one of A or B
Intersection of A and B
A ∩ B
All the outcomes in S that belong to both A and B
STA 291 Spring 2008 Lecture 7

4. Probability Let A and B be two events in a sample space S
P(A∪B)=P(A)+P(B)-P(A∩B)
A and B are Disjoint (mutually exclusive) events if there are no outcomes common to both A and B
A∩B=Ø
Ø = empty set or null set
P(A∪B)=P(A)+P(B)
STA 291 Spring 2008 Lecture 7

5. Assigning Probabilities to Events
Can be difficult
Different approaches to assigning probabilities to events
Objective
Equally likely outcomes (classical approach)
Relative frequency
Subjective
STA 291 Spring 2008 Lecture 7

6. Equally Likely Approach
The equally likely outcomes approach usually relies on symmetry to assign probabilities to events
As such, previous research or experiments are not needed to determine the probabilities
Suppose that an experiment has only n outcomes
The equally likely approach to probability assigns a probability of 1/n to each of the outcomes
Further, if an event A is made up of m outcomes, then
P(A) = m/n
STA 291 Spring 2008 Lecture 7

7. Relative Frequency Approach
Borrows from calculus’ concept of the limit
We cannot repeat an experiment infinitely many times so instead we use a ‘large’ n
Process
Repeat an experiment n times
Record the number of times an event A occurs, denote this value by a
Calculate the value of a/n
STA 291 Spring 2008 Lecture 7

8. Subjective Probability Approach
Relies on a person to make a judgment as to how likely an event will occur
Events of interest are usually events that cannot be replicated easily or cannot be modeled with the equally likely outcomes approach
As such, these values will most likely vary from person to person
The only rule for a subjective probability is that the probability of the event must be a value in the interval [0,1]
STA 291 Spring 2008 Lecture 7

9. Probabilities of Events
Let A be the event A = {o1, o2, …, ok}, where o1, o2, …, ok are k different outcomes
Suppose the first digit of a license plate is randomly selected between 0 and 9
What is the probability that the digit 3?
What is the probability that the digit is less than 4?
STA 291 Spring 2008 Lecture 7

10. Conditional Probability
Note: P(A|B) is read as “the probability that A occurs given that B has occurred”
STA 291 Spring 2008 Lecture 7

11. Independence
If events A and B are independent, then the events have no influence on each other
P(A) is unaffected by whether or not B has occurred
Mathematically, if A is independent of B
P(A|B)=P(A)
Multiplication rule for independent events A and B
P(A∩B)=P(A)P(B)
STA 291 Spring 2008 Lecture 7

12. Example
Flip a coin twice, what is the probability of observing two heads?
Flip a coin twice, what is the probability of observing a head then a tail? A tail then a head? One head and one tail?
A 78% free throw shooter is fouled while shooting a three pointer, what is the probability he makes all 3 free throws? None?
STA 291 Spring 2008 Lecture 7

13. Random Variables
X is a random variable if the value that X will assume cannot be predicted with certainty
That’s why its called random
Two types of random variables
Discrete
Can only assume a finite or countably infinite number of different values
Continuous
Can assume all the values in some interval
STA 291 Spring 2008 Lecture 7

14. Examples Are the following random variables discrete or continuous?
X = number of houses sold by a real estate developer per week
X = weight of a child at birth
X = time required to run 800 meters
X = number of heads in ten tosses of a coin
STA 291 Spring 2008 Lecture 7

15. Discrete Probability Distribution
A list of the possible values of a random variable X, say (xi) and the probability associated with each, P(X=xi)
All probabilities must be nonnegative
Probabilities sum to 1
STA 291 Spring 2008 Lecture 7

16. Example X 1 2 3 4 5 6 7
P(X) .1 .2
.15
.05
The table above gives the proportion of employees who use X number of sick days in a year
An employee is to be selected at random
Let X = # of days of leave
P(X=2) =
P(X≥4) =
P(X<4) =
P(1≤X≤6) =
STA 291 Spring 2008 Lecture 7

17. Expected Value of a Discrete Random Variable
Expected Value (or mean) of a random variable X
Mean = E(X) = μ = ΣxiP(X=xi)
Example
E(X) = X 2 4 6 8 10 12
P(X) .1
.05 .4
.25
STA 291 Spring 2008 Lecture 7

18. Variance of a Discrete Random Variable
Var(X) = E(X-μ)2 = σ2 = Σ(xi-μ)2P(X=xi)
Example
Var(X) = X 2 4 6 8 10 12
P(X) .1
.05 .4
.25
STA 291 Spring 2008 Lecture 7

19. Bernoulli Random Variables
A random variable X is called a Bernoulli r.v. if X can only take either the value 0 (failure) or 1 (success)
Heads/Tails
Live/Die
Defective/Nondefective
Probabilities are denoted by
P(success) = P(1) = p
P(failure) = P(0) = 1-p = q
Expected value of a Bernoulli r.v. = p
Variance = pq
STA 291 Spring 2008 Lecture 7

20. Binomial Distribution
Suppose we perform several, we’ll say n, Bernoulli experiments and they are all independent of each other (meaning the outcome of one even doesn’t effect the outcome of another)
Label these n Bernoulli random variables in this manner: X1, X2,…,Xn
The probability of success in a single trial is p
The probability of success doesn’t change from trial to trial
We will build a new random variable X using all of these Bernoulli random variables:
What are the possible outcomes of X? What is X counting?
STA 291 Spring 2008 Lecture 7

21. Binomial Distribution
The probability of observing k successes in n independent trails is
Assuming the probability of success is p
Note:
Why do we need this?
STA 291 Spring 2008 Lecture 7

22. Binomial Coefficient
For small n, the Binomial coefficient “n choose k” can be derived without much mathematics
STA 291 Spring 2008 Lecture 7

23. Example Assume Zolton is a 68% free throw shooter
What is the probability of Zolton making 5 out of 6 free throws?
What is the probability of Zolton making 4 out of 6 free throws?
STA 291 Spring 2008 Lecture 7

24. Binomial Distribution Properties
STA 291 Spring 2008 Lecture 7