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(A 1 or A 2 )= P(A 1 ) + P (A 2 ) if A 1 and A 2 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(A i |B j ) = P(A i ) 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 { x 1, x 2, x 3, …..x n }, then the expected value of x is simply the sum of x 1 through x n, –The expected value of x also is referred to as average or mean of x, E(x) or occasionally µ x (Greek mu)

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