Chapter 7

1. Chapter 7 • Probability Basics

2. Chapter 7 • Introduction to Probability Basics • Learning Objectives • Probability Theory and Concepts • Use of probability to model simple uncertain situation • Discrete probability Distribution • Continuous Probability Distribution

3. Chapter 7 (Probability Basics) • Probabilities must satisfy the following: • 1) Probabilities must lie between 0 and 1 (0<=P<=1) • 2) Probabilities must add up P(A1 or A2)= P(A1) + P (A2) if A1 and A2 can’t both happen • 3) Total probability must equal 1

4. Probability Basics • Venn Diagrams • More Probability Formulas • Conditional Probability • P(A|B)=P(A and B)/ P(B) • Independence • P(Ai|Bj) = P(Ai) • Complements • P(B) = 1-P(B)

5. Probability Basics • Total Probability of an Event • P(A) =P(A and B)+P(A and B) • =P(A|B) P(B) + P(A|B) P(B) • Venn Diagram for total probability of an event

6. Probability Basics • Bayes’ Theorem : • P(B|A) = P(A|B) P(B) / P(A|B) P(B) • Probability Distribution • Random Variable • Use of capital letter to present uncertain quantities

7. Discrete Probability Distribution • The discrete probability distribution characteristic • Examples of discrete probability distribution: • Number of raisins in an oatmeal cookie, number of operations a computer performs in any given second, number of games that will be won by the Los Angeles Lakers this season.

8. Discrete Probability Distribution • Probability Mass Function : • Definition of mass function • Example of mass function: No cookie in a batch of oatmeal cookies could have more than five raisins • P(Y=0 raisins)= 0.02 ,P(Y=3 raisins)= 0.40 • P(Y=1 raisin)= 0.05 ,P(Y=4 raisins)= 0.22 • P(Y=2 raisins)= 0.20 ,P(Y=5 raisins)= 0.11

9. Discrete Probability Distribution • The second way to express a discrete probability distribution • Cumulative Distribution Function (CDF) • Example of cumulative distribution function: • P(Y<=0 raisins)= 0.02 , P(Y<=3 raisins)= 0.67 • P(Y<=1 raisin)= 0.07 ,P(Y<=4 raisins)= 0.89 • P(Y<=2 raisins)= 0.27 ,P(Y<=5 raisins)= 1.00

10. Discrete Probability Distribution • Expected Value : is the probability-weighted average of its possible values. • If x can take on any value in the set { x1, x2, x3, …..xn}, then the expected value of x is simply the sum of x1 through xn, • The expected value of x also is referred to as average or mean of x, E(x) or occasionally µx (Greek mu)

11. Discrete Probability Distribution • Variance and standard Deviation: The variance of uncertain quantity X is denoted by Var(X) or 2x (Greek sigma): • Var(X) = E[X-E(X)]2 • In words, calculate the difference between the expected value and xi and square that difference.

12. Discrete Probability Distribution • The Standard Deviation: • Square root of the variance x • Standard Deviation as a “best guess” • Use of variance and standard deviation of a probability distribution as measures of variability

13. Continuous Probability Distributions • Continuous Probability Distribution • Examples of continuous probability distribution: • Temperature tomorrow at LAX at noon that can be anywhere between 65°F and 100ºF. • The length of time until some anticipated event ( the next major earthquake in CA)

14. Continuous Probability Distributions • Cumulative Distribution Function (CDF) in Continuous Probability Distribution • Example of CDF: movie star’s age • She is older than 29 and no older than 65 • P(Age<=29)=0 and P(Age<=65)=1 • She is most likely is between 40 and 50 years old, and that P(40<Age<=50)=0.8 • P(Age<=40)=0.05 & P (Age>50)=0.15

15. Continuous Probability Distribution • Probability Density Function f(x) • Example of Probability Density Function • Expected value, Variance and Standard Deviation in Continuous Case • Integration of density function is a difficult task • Use of formulas for the expected values and variances

16. Chapter 7, Probability Basics • Summary • Use of probability for uncertain situation in decision analysis • Venn diagram provide visualization • Discrete Probability Distribution • Continuous Probability Distribution