Supplement to Chapter 2 - Probability and Statistics
Learning Objectives
After completing this chapter you should be able to:
- Define and explain the terms event, mutually exclusive, sample space and Venn diagram
- Describe the three different approaches toassigning probabilities:
classical,relative frequency, and subjective.
- Explain what is meant by marginal, conditional, and joint probability.
- Contrast independent and dependent events.
- Calculate variance, standard deviation and coefficient of variation for discrete probability distributions and samples.
- Compare and distinguish between discrete and continuous random variables with examples.
- List the assumptions and solve problems using binomial distribution.
- State the parameters and uses of normal distribution.
- Use the Z table to find the probability of a normally distributed random variable X
Chapter Outline
Introduction, 2
Sample space and events, 2
Concepts of Probability, 2
Probability Axioms, 3
Unions and Intersections, 3
Venn Diagrams, 3
Methods of Assigning Probabilities, 5
Classical Approach, 5
Relative Frequency Approach, 6
Subjective Approach, 6
Conditional Probability, 6
Joint Probability, 7
Marginal Probability, 7
Computing the Probability that Both Events Will Occur, 7
Random Variables, 9
Expected Value, Variance, and Standard Deviation of a Discrete
Random Variable, 9
Sample Mean and Sample Variance, 12
Coefficient of Variation, 14
Binomial Distribution, 14
Normal Distribution, 16
Introduction
The techniques and methods covered in this chapter not only will be useful in covering the material in chapter 2, but also covering the material in the probabilistic models section in chapters 11, 12, 13 and 14. Probabilities express the degree ofcertaintyin managerial decision making or any situation involving uncertainty. Assigning probabilities to future events allows us to analyze decision options in a rational way. For instance, apersonis more likely to carry an umbrella if the probability of rain is highthan when the probabilityofrain is low. The ability to quantify the likelihood of the occurrence of some future event well allows us to make sound decisions
Sample Space and Events
When we specify all possible outcomes in an experiment or a study, we are stating the sample space.Asample space is the collection of all the experimental outcomes. Examples of experimental outcomes maybe the potential outcome of a soccer game (win, loose or tie). For a soccer game result we just listed the sample space since our list above included all possible outcomes. If the experiment involves rolling a fair die, the sample space or the experimental outcomes are (1, 2, 3, 4, 5, and 6). Since all of the possible outcomes are included, this constitutes a sample space.
An event is anysubset of asamplespace. It consists of one or more outcomes with a common characteristic, such as even numbered outcomes for a single roll of die.
When a die is rolled, sample space consists of 6 outcomes i.e. S = (1, 2, 3, 4, 5, and 6). When a local college classifies its students as S = (deposit paid, deposit unpaid). The sample space describing the method of payment at the local grocery store might have four outcomes: S = (cash, credit card, check, debit card). The term experimentsuggests that the outcome is uncertain prior to taking the observations. If an experiment consists of outcomes generated by a sequential process such a flipping a coin three times and recording whether it is heads or tails can be represented as a tree diagram which is illustrated figure 2S-1. On each flip either a head or tail will occur. The final sequence of heads and tails will depend on the outcome of each flip. In figure 2S-1, we can see that
There are eight possible outcomes in this experiment.
Concepts of Probability
Probabilities are stated with reference to some event. The event in question might be rain, winning a game, and head on single flip of a coin(50% chance). All of the above examplesaresituations that could be considered a random experiment. A random experiment is a process of observation for which the outcome is uncertain.
Probability is a measure of likelihood of the occurrence of an event. Possible values range from 0% to 100%. The closer a probability is to 100%, the more likely it is to occur. Probabilities are often expressed as decimals. For example, if the probability of outcome A is 75%, we would express that as P(A) = .75.There are other ways of expressing probabilities. For instance .75 is a decimal expression of probability of A, and 3 out of 4 is a fraction expression of probability of A. Since a sample space consists of all the possible outcomes of an experiment, the probability of a sample space is 100 %.
In doing the probability calculations, we have to consider concepts of a complimentof an event, mutually exclusiveevents,and collectively exhaustiveevents. The complement of an event consists of all outcomes of the event that are not part of the original event A. For example, the probability of complement of event A is symbolized by, which is the probability of everything that is not in A. Mutually exclusive events are signified by the fact that two events cannot occur jointly or simultaneously. Therefore if two events A and B have some component in common, we can conclude that event A and B are not mutually exclusive.Events are said to be collectively exhaustiveif the comprise the entire sample set; no other events are possible.
Probability Axioms
All probability statements must satisfy the following two probability axioms:
- The probability of event A must be a value between zero and1
.
2If Ai i = 1,2, …, j isa set of mutually exclusive and collectively exhaustive events in the sample space S, then the sum of the probability of these events is1.00. That is:, where j is the number of events in S.
Unions and Intersections
Set theory can be instrumental in explaining a few simple concepts. Let’s consider the terms union and intersection. The union of two events A and B consists of all outcomes belonging to either or both events A or B. In the set notation the union of A and B is A B. The key term in union is or. The intersection of two events A and B consists of the outcomes that belong to event A and event B. The outcome must belong to both events to be an intersection. In the set notation, the intersection of event A and event B is A B. The key term in intersection is and. In other words “both” implies “and” which implies intersection,and “either” implies “or” which implies union.
For events that are not mutually exclusive: .
For mutually exclusive events:
Venn Diagrams
It can be helpfulto show the sample space in a graphical form, because the graph makes it easier to understand the relationships between events. This visualization can be accomplished by using a Venn diagram, which depicts mutually exclusive and non mutually exclusive events. Read Example 2S-1and then look at the Venn diagram shown in figure 2S-2.
Example 2S-1
The school book store orders two colors of glasses (red and green) with the school logo. The colored glasses are packed randomly. 40% of the glasses are red. There is a 20% chance that the glass will be damaged during shipment and a 8% chance that it will be a damaged red glass. What is the probability that a glass that is randomly selected:
a)Will be red or damaged?
b)Will be green?
c)Will not be red or damaged?
d)Draw a Venn diagram for Exercise 2S-1part a.
Solution
R= Red G = Green D= Damaged and
a)
b) P(G)= 1 - P(R)=1 - .4 = .6
c)
d)Since P(R) = .40, P(D) =.2 , giving us the followingVenn diagram.
Figure 2S-1
Exercises
1.The personnel manager of a company has gathered information about the company. This information is displayed in Figure 2S-2. One employee is to be selected at random.
Figure 2S-2
N = Neither degree
A = Engineering degree
B = Business degree
a)What is the probability that the employee has an engineering degree?
b) What is the probability that the employee has a business degree?
c) What is the probability that an employee has at least one of these degrees?
d) What is the probability that the selected employee has neither degree?
Solution
a)Total = 125 + 310 +500 = 935
A/Total = 125/935 = .1337
b)B/Total = 310/935 = .3316
c)(A + B)/Total = (125 + 310)/935 = 435/935 = .4652
d)N/Total = 500 / 935 = .5348
Methods of Assigning Probabilities
There are three methods of assigning probabilities:
- The classical approach
- The relative frequency approach
- The subjective approach
The Classical Approach
This approach applies when the events have equally likely outcomes. Games of chance such as tossing a coin or rolling a die are based on theclassical approach. Let’s take the example of rolling a die. Since a die has six possible outcomes, the probability of any outcome on a fair die based on a classical approach is 1/6. Likewise, the probability of head on a toss of a fair coin is ½.
The Relative Frequency Approach
The classical approach to assignment of probabilities involves situations in which outcomes are equally likely. As you might guess the probability of outcomes are not equally likelyin many circumstances. For example, we may have surveyed 200 people about a taste test of three brands of soft drinks: Pepsi-Cola, Coca-Cola, and RCCola.Suppose that 90 out of 200 preferred Pepsi-Cola. In other words P (Pepsi) = 90 / 200 = .45 or 45%. Hence according to the relative frequency approach, we have the following definition
The Subjective Approach
Probabilities computed by either theclassical approach orthe relative frequency approach are called objective probabilities because these probabilities are based on objective facts.
However, there are numerous situations that don’t lend themselves to an objective approach. In situations in which the outcomes are not equally likely or historical data is not readily available a subjective probability must be assigned. For example the probability that a particular new product will be a success is a subjective
assessment of probability.Since this type of assessment is based on the personal opinionof the decision maker, subjective probability is defined as follows: Subjective Probability
is a personal assessment of the likelihood of an event. Therefore subjective probability involves our efforts to quantify our feeling or beliefs about something, such as the chance of success of a new product. Disadvantages ofsubjective probabilityare:
- Difficult to support if questioned
- Biases can be a factor. Preconceived notions about what should happen can distort objectivity. It is often difficult to eliminate biases because they are often subconscious.
Conditional Probability
ConditionalProbability is the likelihood that an event will occur given that some other event is already knownto have occurred. The notation used is which is read as the probability of A given B. implies that B is already known to have occurred.
In general, we can compute this probability by using:
Example 2S-2 A company that does tax returns has determined that the probability that a tax return will require a refund is 55%. The probability that a tax return will require a refund and the return contains an error is 20%. A tax return has been selected and known to require a refund Find the probability that the selected tax return has an error given that it requires a refund.
Solution
P(R) = .55 (probability of refund required)
(probability of refund required and has an error)
We can find the conditional probability of an error given a refund.
Joint ProbabilityA joint probability is the probability that two or more events occur simultaneously or sequentially. A joint probability is indicated by the word “and”. The word and signifies the intersection of two events. This is signified by
. In the caseof simultaneous occurrence of two events:
Marginal Probability Given a sample space S and event A, the probability that A will occurs is called a marginal probability. This is denoted by P(A).
Example 2S-3 Assume that 30 out of 50 employees at a manufacturing company are production workers. 12 of the 30 employees are between ages 18-29, 10 of the 30 employees are between ages 30 to 49 and 8 of the 30 employees are between ages 50 to 65. What is the probability that a single selected employee is a production worker?
Solution
Let P = Production worker
A1 = The worker is between ages 18 to 29
A2 = The worker is between ages 30 to 49
A3 = The worker is between ages 50 to 65
There are two ways to do this problem. The first way does not require the knowledge of age breakdown:
The second way does not require us to know the overall number of production workers but does require usto know the number of production workers at different age levels for all age levels.
Thus,we obtain the same answer with both methods. Note that it is permissible to add the joint probabilities because the three events are mutually exclusive and probability axiom 2 is satisfied.
Computing the Probability That Two Events Will Both Occur:
The probability that both events will occur simultaneously is called their joint probability and its computationdepends onwhether the events in question are dependent or independent.
Two events are considered to be independent of each other if the occurrence of one event is unrelated to the occurrence of the other. For instance, if two fair dice are rolled, knowing the result of one die does not aid in determining the result of the other die.
Two events are considered to be dependent if one event occurring affects the probability of the other occurring. For example, knowing that a potential buyer has seen a newspaper advertisement about a product, there is a higher probability of making a sale of that product.
The probability that two or more events will occur simultaneously or sequentially iscalled a joint probability. Itcan be computed using themultiplication rule. A joint probability is indicated by the word “and”, it signifies the intersection of two events.If two events are dependent, the probability that both will occur simultaneously is equal to the probability that the first one occurs times the conditional probability of the second one occurring.
Example 2S-2 Let’s consider an example with two dependent events A and B. Let’s also assume that event A occurs first. What is the formula of joint probability of event A and event B?
Solution
Example 2S-3 Let’s consider an example with two independent events A and B. Let’s also assume that event A occurs first. What is the joint probability of event A and event B?
Solution
Let’s consider the following two examples to illustrate joint probability
Example 2S-4 Two consumers are to be selected at random from a group of five consumers (A, B, C, D, E) to be interviewed by a product representative. What is the probability that both C and E will be interviewed?
Solution Since event C and event E are dependent, we can employ the dependent event formula for joint probability:
Example 2S-5 One-third of the consumers of a certain deodorant are women,and 70% of consumers chose a certaindeodorant because of its nice smell. Find the probability that the next randomly selected consumer will be a woman and will have selected this product because of its nice smell. Assuming these two events are independent we can employ the multiplication rule for independent events:
Let W =Women and S = Select because of its nice smell
P(W) = .3333 P(S)=.70
Solution
Random Variables
A random variable is a function that assigns numbers to the basic experimental outcomes. Let’s consider the coin example and let X(tail) = 0 and X(head) = 1 in which case the variable X is defined as the number of heads occurring.
Generally speaking random variables areeitherdiscrete or continuous. It is important to distinguish between discrete and continuous random variables because different mathematical techniques are utilized depending on which type is involved.A random variable is said to be discrete if it can assume only a finite number of values. A random variable is said to be continuous if it can assume any value in some interval or set of intervals. In the continuous case, the set of possible outcomes is always infinite. Therefore, it is not possible to list the sample space by individual values in any form. The way to distinguish between discrete and continuous is to ask whether the values of the random variable can be counted. Theoutcomes ofdiscrete random variables can be counted (e.g.,the number of heads in 10 coin tosses number, or the number of defectivesitems in batch of 300 units). The outcomes ofcontinuous random variables are measured rather than counted (e.g., theweight of an individual,or thelength of a bolt).
Expected Value, Variance and Standard Deviation of a Discrete Random Variable
In working with discrete random variables it is helpful to know certain values that aid in describing the distribution. The most commonly used values are those that identify the physical centerand the dispersion (the way the values are spread around the center). Given a discrete random variable X, with probability function p(x) the expected value of X(also called the mean and denoted E[X] ) or (mean of random variable X) is defined as the weighted average of the values of X may assume where weights are the corresponding probabilities, that is:
Example 2S-6
Determine the expected number of broken tools per day for the probability distribution given in Table 2S-1:
Table 2S-1 Discrete Probability Distribution
for Broken Tools
Number Broken per day / P(X) / X P(X)0 / .23 / 0
1 / .50 / .50
2 / .15 / .30
3 / .08 / .24
4 / .04 / .16
Total / 1.0 / 1.2
Hence the mean or expected number of broken tools per day is 1.20. This value can be interpreted as the long run average of broken tools per day. Obviously, this value cannot occur on any day since the average is not an integer value. However in interpreting this value say over 50 days, the factory can expect to have (50).(1.2)= 60 broken tools. This result does not imply that exactly 60 tools will be broken over the 50 day period, but it does provide management withan estimate of the replacement toolsthat will be needed. This figure (60 tools) can not only be used to estimate the number of replacements, it can alsobe used to for other planning purposes such as estimating the cost of the replacements, estimating the downtime due to tool breakage, etc.
It would also be useful for a decision maker to have some knowledge about the extent to which the actual number of breakages might tend to vary from the expected value of breakages by calculating the variance or the standard deviation.