Using Probability and Discrete Probability Distributions Chapter 4 Using Probability and Discrete Probability Distributions

Chapter 4 - Chapter Outcomes After studying the material in this chapter, you should be able to: • Understand the three approaches to assessing probabilities. • Apply the common rules of probability. • Identify the types of processes that are represented by discrete probability distributions.

Chapter 4 - Chapter Outcomes (continued) After studying the material in this chapter, you should be able to: • Know how to determine probabilities associated with binomial and Poisson distribution applications.

Probability refers to the chance that a particular event will occur. The probability of an event will be a value in the range 0.00 to 1.00. A value of 0.00 means the event will not occur. A probability of 1.00 means the event will occur. Anything between 0.00 and 1.00 reflects the uncertainty of the event occurring.

Events and Sample Space An experiment is a process that produces a single outcome whose result cannot be predicted with certainty.

Events and Sample Space Elementary events are the most rudimentary outcomes resulting from a simple experiment.

Events and Sample Space The sample space is the collection of all elementary outcomes that can result from a selection or decision.

Events and Sample Space An event is a collection of elementary events.

Events and Sample Space (Able Accounting Example) Sample Space = SS = (e1, e2, e3, e4, e5, e6, e7, e8, e9)

Mutually Exclusive Events Two events are mutually exclusive if the occurrence of one event precludes the occurrence of a second event.

Mutually Exclusive Events (Able Accounting Example) The event in which at least one of the two audits is late: E1 = {e3, e6, e7, e8, e9} The event that neither audit is late: E2 = {e1, e2, e4, e5} E1 and E2 are mutually exclusive!

Independent and Dependent Events Two events are independent if the occurrence of one event in no way influences the probability of the occurrence of the other event.

Independent and Dependent Events Two events are dependent if the occurrence of one event impacts the probability of the other event occurring.

Classical Probability Assessment Classical Probability Assessment refers to the method of determining probability based on the ratio of the number of ways the event of interest can occur to the total number of ways any event can occur when the individual elementary events are equally likely.

Classical Probability Assessment CLASSICAL PROBABILITY MEASUREMENT

Relative Frequency of Occurrence Relative Frequency of Occurrence refers to a method that defines probability as the number of times an event occurs, divided by the total number of times an experiment is performed in a large number of trials.

Relative Frequency of Occurrence where: Ei = the event of interest RF(Ei) = the relative frequency of Ei occurring n = number of trials

Relative Frequency of Occurrence (Example 4-6)

Subjective Probability Assessment Subjective Probability Assessment refers to the method that defines probability of an event as reflecting a decision maker’s state of mind regarding the chances that the particular event will occur.

The Rules of Probability PROBABILITY RULE 1 For any event Ei 0.0  P(Ei)  1.0 for all i

The Rules of Probability PROBABILITY RULE 2 where: k = Number of elementary events in the sample space ei = ith elementary event

The Rules of Probability PROBABILITY RULE 3 The probability of an event Ei is equal to the sum of the probabilities of the elementary events forming Ei. That is, if: Ei = {e1, e2, e3} then: P(Ei) = P(e1) + P(e2) + P(e3)

Complements The complement of an event E is the collection of all possible elementary events not contained in event E. The complement of event E is represented by E.

The Rules of Probability COMPLEMENT RULE

The Rules of Probability PROBABILITY RULE 4 Addition rule for any two events E1 and E2: P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2)

The Rules of Probability PROBABILITY RULE 5 Addition rule for mutually exclusive events E1 and E2: P(E1 or E2) = P(E1) + P(E2)

Conditional Probability Conditional probability refers to the probability that an event will occur given that some other event has already happened.

The Rules of Probability PROBABILITY RULE 6 Conditional probability for any two events E1 , E2:

Tree Diagrams Another way of organizing events of an experiment that aids in the calculation of probabilities is the tree diagram.

Tree Diagrams (Figure 4-1) Male P(E5) = 0.66 Female P(E4) = 0.34

Tree Diagrams (Figure 4-1) P(E1) = 0.38 P(E2) = 0.44 Male P(E5) = 0.66 P(E3) = 0.18 P(E1) = 0.38 Female P(E4) = 0.34 P(E2) = 0.44 P(E3) = 0.18

Tree Diagrams (Figure 4-1) P(E1 and E5) = 0.38 x 0.66 = 0.20 P(E1) = 0.38 P(E2) = 0.44 Male P(E5) = 0.66 P(E2 and E5) = 0.44 x 0.66 = 0.32 P(E3) = 0.18 P(E3 and E5) = 0.18 x 0.66 = 0.14 P(E1 and E4) = 0.38 x 0.34 = 0.18 P(E1) = 0.38 Female P(E4) = 0.34 P(E2) = 0.44 P(E2 and E4) = 0.44 x 0.34 = 0.12 P(E3) = 0.18 P(E3 and E4) = 0.18 x 0.34 = 0.04

The Rules of Probability PROBABILITY RULE 7 Conditional probability for independent events E1 , E2:

The Rules of Probability PROBABILITY RULE 8 Multiplication rule two events E1 and E2:

The Rules of Probability PROBABILITY RULE 9 Multiplication rule independent events E1 , E2:

Bayes’ Theorem BAYES’ THEOREM where: Ei = ith event of interest of the k possible events B = new event that might impact P(Ei)

Discrete Probability Distributions A random variable is a variable that assigns a numerical value to each outcome of a random experiment or trial.

Discrete Probability Distributions A discrete random variable is a variable that can only assume a countable number of values.

Discrete Probability Distributions A continuous random variable is a variable that can assume any value on a continuum. Alternatively, they are random variables that can assume an uncountable number of values.

Discrete Distributions (Example 4-19)

Discrete Distributions (Example 4-19) Discrete Probability Distribution x = Number of service calls

Discrete Probability Distributions The uniform probability distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable

Discrete Distributions (Example 4-20) Uniform Probability Distribution Delivery Lead Time

Mean and Standard Deviation of Discrete Distributions EXPECTED VALUE FOR A DISCRETE DISTRIBUTION where: E(x) = Expected value of the random variable x = Values of the random variable P(x) = Probability of the random variable taking on the value of x

Mean and Standard Deviation of Discrete Distributions STANDARD DEVIATION FOR A DISCRETE DISTRIBUTION where: E(x) = Expected value of the random variable x = Values of the random variable P(x) = Probability of the random variable having the value of x

Binomial Probability Distribution • A manufacturing plant labels items as either defective or acceptable. • A firm bidding for a contract will either get the contract or not. • A marketing research firm receives survey responses of “Yes, I will buy,” or “No, I will not.” • New job applicants either accept the offer or reject it.

Binomial Probability Distribution Characteristics of the Binomial Probability Distribution: • A trial has only two possible outcomes – a success or a failure. • There is a fixed number, n, of identical trials. • The trials of the experiment are independent of each other and randomly generated. • The probability of a success, p, remains constant from trial to trial. • If p represents the probability of a success, then (1-p) = q is the probability of a failure.

Combinations A combination is an outcome of an experiment where x objects are selected from a group of n objects.

COUNTING RULE FOR COMBINATIONS where: n! =n(n - 1)(n - 2) . . . (2)(1) x! = x(x - 1)(x - 2) . . . (2)(1) 0! = 1

Binomial Probability Distribution BINOMIAL FORMULA where: n = sample size x = number of successes n - x = number of failures p = probability of a success q = 1 - p = probability of a failure n! =n(n - 1)(n - 2) . . . (2)(1) x! = x(x - 1)(x - 2) . . . (2)(1) 0! = 1

Binomial Probability Distribution Table

Binomial Probability Distribution MEAN OF THE BINOMIAL DISTRIBUTION where: n = Sample size p = Probability of a success

Binomial Probability Distribution STANDARD DEVIATION FOR THE BINOMIAL DISTRIBUTION where: n = Sample size p = Probability of a success q = (1 - p) = Probability of a failure

Poisson Probability Distribution Characteristics of the Poisson Probability Distribution: • The outcomes of interest are rare relative to the possible outcomes. • The average number of outcomes of interest per segment is .. • The number of outcomes of interest are random, and the occurrence of one outcome does not influence the chances of another outcome of interest. • The probability of that an outcome of interest occurs in a given segment is the same for all segments.

Poisson Probability Distribution where: x = number of successes in segment t t = expected number of successes in segment t e =base of the natural number system (2.71828)

Poisson Probability Distribution Table

Mean and Standard Deviation for the Poisson Probability Distribution MEAN OF THE POISSON DISTRIBUTION STANDARD DEVIATION FOR THE POISSON DISTRIBUTION

Key Terms • Binomial Probability Distribution • Classical Probability • Conditional Probability • Continuous Random Variable • Dependent Events • Discrete Random Variable • Elementary Events • Event • Independent Events • Mutually Exclusive Events • Poisson Probability Distribution • Probability • Random Variable

Key Terms (continued) • Relative Frequency of Occurrence • Sample Space • Subjective Probability Assessment • Uniform Probability Distribution