Chapter 2 Probability Concepts and Applications

Objectives Students will be able to: Understand the basic foundations of probability analysis Do basic statistical analysis Know various type of probability distributions and know when to use them

Probability Life is uncertain and full of surprise. Do you know what happen tomorrow Make decision and live with the consequence The probability of an event is a numerical value that measures the likelihood that the event can occur

Basic Probability Properties Let P(A) be the probability of the event A, then The sum of the probability of all possible outcomes should be 1.

Mutually Exclusive Events Two events are mutually exclusive if they can not occur at the same time. Which are mutually exclusive? Draw an Ace and draw a heart from a standard deck of 52 cards It is raining and I show up for class Dr. Li is an easy teacher and I fail the class Dr. Beaubouef is a hard teacher and I ace the class.

Addition Rule of Probability If two events A and B are mutually exclusive, then Otherwise

P(A or B) + - = P(A) P(B) P(A and B) P(A or B)

Independent and Dependent Events are either statistically independent (the occurrence of one event has no effect on the probability of occurrence of the other) or statistically dependent (the occurrence of one event gives information about the occurrence of the other)

Which Are Independent? (a) Your education (b) Your income level (a) Draw a Jack of Hearts from a full 52 card deck (b) Draw a Jack of Clubs from a full 52 card deck (a) Chicago Cubs win the National League pennant (b) Chicago Cubs win the World Series

Conditional Probability the probability of event B given that event A has occurred P(B|A) or, the probability of event A given that event B has occurred P(A|B)

Multiplication Rule of Probability If two events A and B are mutually exclusive, Otherwise,

Joint Probabilities, Dependent Events Your stockbroker informs you that if the stock market reaches the 10,500 point level by January, there is a 70% probability the Tubeless Electronics will go up in value. Your own feeling is that there is only a 40% chance of the market reaching 10,500 by January. What is the probability that both the stock market will reach 10,500 points, and the price of Tubeless will go up in value?

Probability(A|B) / P(A|B) = P(AB)/P(B) P(AB) P(B) P(A)

Random Variables Discrete random variable - can assume only a finite or limited set of values- i.e., the number of automobiles sold in a year Continuous random variable - can assume any one of an infinite set of values - i.e., temperature, product lifetime

Random Variables (Numeric) Experiment Outcome Random Variable Range of Random Variable Stock 50 Xmas trees Number of trees sold X = number of 0,1,2,, 50 Inspect 600 items Number acceptable Y = number 0,1,2,…, 600 Send out 5,000 sales letters people e responding Z = number of people responding 5,000 Build an apartment building %completed after 4 months R = %completed after 4 months £ R 100 Test the lifetime of a light bulb (minutes) Time bulb lasts - up to 80,000 minutes S = time bulb burns S

Probability Distributions Figure 2.5 Probability Function

Expected Value of a Discrete Probability Distribution å = n i ) X ( P E 1

Variance of a Discrete Probability Distribution

Binomial Distribution Assumptions: 1. Trials follow Bernoulli process – two possible outcomes 2. Probabilities stay the same from one trial to the next 3. Trials are statistically independent 4. Number of trials is a positive integer

Binomial Distribution n = number of trials r = number of successes p = probability of success q = probability of failure Probability of r successes in n trials

Binomial Distribution

Binomial Distribution N = 5, p = 0.50

Probability Distribution Continuous Random Variable Normal Distribution Probability density function - f(X)

Normal Distribution for Different Values of  =50 =60 =40

Normal Distribution for Different Values of   = 1 =0.1 =0.3 =0.2

Three Common Areas Under the Curve Three Normal distributions with different areas

Three Common Areas Under the Curve Three Normal distributions with different areas

The Relationship Between Z and X =100 =15

Haynes Construction Company Example Fig. 2.12

Haynes Construction Company Example Fig. 2.13

Haynes Construction Company Example Fig. 2.14

The Negative Exponential Distribution Expected value = 1/ Variance = 1/2 =5

The Poisson Distribution Expected value =  Variance =  =2