Chapter 22

1. Chapter 22 Probability

2. Objectives • Describe and apply probability concepts such as independence, mutually exclusive events, joint occurrence of events, etc. • Describe, apply, and interpret distributions: Poisson, binomial, chi square, t and F distributions.

3. Basic Concepts • Probability (definition): If an event, say A, can occur in m ways out of a possible n equally likely ways, then: P(A) = m / n • The probability that a particular event occurs is a number between 0 and 1. • For example, if a lot consisting of 100 parts has four defectives, we would say the probability of randomly drawing a defective is 4/100 = 0.04 or 4%.

4. Rules of Probability • The complementary rule of probability states that the probability that event A will occur is 1 – P(event A does not occur). • Stated symbolically, P(A) = 1 – P (not A). • Addition rule of probability: • P(A or B) = P(A) + P(B) • Using the proper symbology: • P(A U B) = P(A) + P(B) • Where U is the symbol for union or addition. • This formula applies only when A and B are independent of each other (fig 22.3, page 140).

5. Contingency Tables • See table 22.2, page 142 for table and for examples on how to use the table. • Conditional Probability: • The probability of B occurring given A occurred (page 143): • P (B|A) = P (A and B) / P (A) = P (A Ω B) / P (A) • Independent events: • If the occurrence of B has no effect on the occurrence of A and vice versa: • P (A|B) = P (A) • P (B|A) = P (B) • Similarly events A and B are dependent if and only of the above formula does not hold. • Multiplication rule: • P (A Ω B) = P (A) X P (B|A) • P (A Ω B) = P (B) X P (A|B)

6. Commonly Used Distributions (table 22.4, page 149) • 1. Normal Distribution: If X is a normal random variable with expected value μ (E(X) = μ ) and finite variance σ2 , then the random variable Z is defined as Z = (X - μ) / σ • See example 22.14 on page 150. • Poisson Distribution: It is important because it is used to model the number of defects per unit of time (or distance). The formula is P (X = x) = (e -λλx)/ x ! For λ > 0 and x = 0, 1, 2,… • See example 22.16 on page 152

7. Commonly Used Distributions • 3. Binomial Distribution: The binomial formula is P(X = x) = (n! / x! (n – x) ! ) p x (1- p) n-x where x! (x factorial) = x (x-1) (x-2)…. (2) (1). Also 0! = 1 by definition and n = sample size or number of trials, X is a random variable, x = number of successes in the sample, and p = probability of success for each trial. • See example 22.17 on page 154.

8. Commonly Used Distributions • 4. Chi-Square (χ 2) Distribution: • = (n-1)s 2 / σ2 • Where we obtain a random sample X1, X2, …Xn of size n = sample size with a sample variance s 2 ,population mean μ and population variance σ2 • 5. t Distribution: • T = (x bar – μ) / (s / √n) • 6. F distribution: • F = (X/ v1) / (Y/ v2) • Where X and Y are two random variables distributed as χ 2 with v1 and v2 degrees of freedom.

9. Summary • The probability that a particular event occurs is a number between 0 and 1. • The complementary rule of probability states that the probability that event A will occur is 1 – P(event A does not occur). • Conditional Probability: The probability of B occurring given A occurred: P (B|A) = P (A and B) / P (A) = P (A Ω B) / P (A) • Independent events: • If the occurrence of B has no effect on the occurrence of A and vice versa: • P (A|B) = P (A) • P (B|A) = P (B) • Similarly events A and B are dependent if and only of the above formula does not hold. • Poisson Distribution: It is important because it is used to model the number of defects per unit of time (or distance). The formula is P (X = x) = (e -λλx)/ x ! For λ > 0 and x = 0, 1, 2,…

10. Home Work • 1. Define probability. • 2. What is the formula for conditional probability? • 3. What is the formula for Poisson distribution and why is it important?