GS ECO 302 CH 5 A Survey of Probability Concepts
Three key words are used in the study of probability:
• experiment, • outcome, • and event.
Approaches to Assigning Probabilities Two approaches to assigning probabilities to an event will be discussed, namely the objective and the subjective viewpoints. Objective probability is subdivided into
(1) classical probability and (2) empirical probability.
If the set of events is collectively exhaustive and the events are mutually exclusive, the sum of the probabilities is
1. It is unnecessary to do an experiment to determine the probability of an event occurring using the classical approach because the total number of outcomes is known before the experiment.
Conditional Probability the probability that one event happens given that another event is already known to have happened the probability of an event (A), given that another (B) has already occurred. If two events are not independent, they are referred to as dependent. To illustrate dependency, suppose there are 10 cans of soda in a cooler, 7 are regular and 3 are diet.
A can is selected from the cooler. The probability of selecting a can of diet soda is 3/10, and the probability of selecting a can of regular soda is 7/10. Then a second can is selected from the cooler, without returning the first. The probability the second is diet depends on whether the first one selected was diet or not. The probability that the second is diet is: • 2/9, if the first can is diet. (Only two cans of diet soda remain in the cooler.) • 3/9, if the first can selected is regular. (All three diet sodas are still in the cooler.)
Event
A collection of one or more outcomes of an experiment
Summary : Types of Probability
A probability statement always assigns a likelihood to an event that has not yet occurred. There is, of course, a considerable latitude in the degree of uncertainty that surrounds this probability, based primarily on the knowledge possessed by the individual concerning the underlying process.
experiment
A process that leads to the occurrence of one and only one of several possible observations. In reference to probability, an experiment has two or more possible results, and it is uncertain which will occur.
contingency table
A table used to classify sample observations according to two or more identifiable categories or classes.
Probability AKA chance, and likelihood
A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur. A probability is frequently expressed as a decimal, such as .70, .27, or .50. However, it may be given as a fraction such as 7/10, 27/100, or 1/2. It can assume any number from 0 to 1, inclusive. Thus, the probability of 1 represents something that is certain to happen, and the probability of 0 represents something that cannot happen. The closer a probability is to 0, the more improbable it is the event will happen. The closer the probability is to 1, the more sure we are it will happen.
Special Rule of Multiplication A rule used to find the probability of the joint occurrence of independent events. P(A and B) = P(A)P(B)
Ex. A survey by the American Automobile Association (AAA) revealed 60 percent of its members made airline reservations last year. Two members are selected at random. What is the probability both made airline reservations last year? The probability the first member made an airline reservation last year is .60, written P(R1)= .60, where R1 refers to the fact that the first member made a reservation. The probability that the second member selected made a reservation is also .60, so P(R2) =.60. Since the number of AAA members is very large, you may assume that R1 and R2 are independent. Consequently, using formula (5-5), the probability they both make a reservation is .36, found by: P(R1 and R2) = P(R1)P(R2) = (.60)(.60) = .36 With the probabilities and the complement rule, we can compute the joint probability of each outcome. For example, the probability that neither member makes a reservation is .16. See Table You can also observe that the outcomes are mutually exclusive and collectively exhaustive. Therefore, the probabilities sum to 1.00.
Law of Large Numbers (LLN) Over a large number of trials, the empirical probability of an event will approach its true probability.
Ex. To explain the law of large numbers, suppose we toss a fair coin. The result of each toss is either a head or a tail. With just one toss of the coin the empirical probability for heads is either zero or one. If we toss the coin a great number of times, the probability of the outcome of heads will approach .5.
If we compare the general and special rules of addition, the important difference is determining if the events are mutually exclusive. If the events are mutually exclusive, then the joint probability P(A and B) is 0 and we could use the special rule of addition. Otherwise, we must account for the joint probability and use the general rule of addition.
Ex. What is the probability that a card chosen at random from a standard deck of cards will be either a king or a heart? We may be inclined to add the probability of a king and the probability of a heart. But this creates a problem. If we do that, the king of hearts is counted with the kings and also with the hearts. So, if we simply add the probability of a king (there are 4 in a deck of 52 cards) to the probability of a heart (there are 13 in a deck of 52 cards) and report that 17 out of 52 cards meet the requirement, we have counted the king of hearts twice. P(A or B) = P(A) + P(B) - P(A and B) = 4/52 (4 kings) + 13/52 (# of hearts) - 1/52 (1 king of hearts) = 16/52
General Rule of Addition The outcomes of an experiment may not be mutually exclusive.
If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula: P(A or B) = P(A) + P(B) - P(A and B) For the expression P(A or B), the word or suggests that A may occur or B may occur. This also includes the possibility that A and B may occur. This use of or is sometimes called an inclusive. You could also write P(A or B or both) to emphasize that the union of the events includes the intersection of A and B.
Special Rule of Addition
If two events A and B are mutually exclusive, the probability of one or the other event's occurring equals the sum of their probabilities P(A or B) = P(A) + P(B) • events must be mutually exclusive Recall that mutually exclusive means that when one event occurs, none of the other events can occur at the same time.
Learning Objectives When you have completed this chapter, you will be able to:
LO1 Explain the terms experiment, event, and outcome. LO2 Identify and apply the appropriate approach to assigning probabilities. LO3 Calculate probabilities using the rules of addition. LO4 Define the term joint probability. LO5 Calculate probabilities using the rules of multiplication. LO6 Define the term conditional probability. LO7 Compute probabilities using a contingency table. LO8 Calculate probabilities using Bayes' theorem. LO9 Determine the number of outcomes using the appropriate principle of counting.
The classical approach to probability can also be applied to lotteries. In South Carolina, one of the games of the Education Lottery is "Pick 3." A person buys a lottery ticket and selects three numbers between 0 and 9.
One way to win is to match the numbers and the order of the numbers. Given that 1,000 possible outcomes exist (000 through 999), the probability of winning with any three-digit number is 0.001, or 1 in 1,000.
Rules of Multiplication When we used the rules of addition in the previous section, we found the likelihood of combining two events. In this section, we find the likelihood that two events both happen.
P(A and B) Venn diagrams illustrate this as the intersection of two events. To find the likelihood of two events happening we use the rules of multiplication. There are two rules of multiplication, • the special rule and • the general rule.
General Rule of Multiplication We use the general rule of multiplication to find the joint probability of two events when the events are not independent. For example, when event B occurs after event A occurs, and A has an effect on the likelihood that event B occurs, then A and B are not independent.
P(A and B) = P(A)P(B|A) The general rule of multiplication states that for two events, A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of event B occurring given that A has occurred. Reads The probability of A and B occurring is = Probability of A * Probability of B after A has occurred
Complement Rule
P(A^c) = 1 - P(~A) Read ~A as "not A" the probability of an event occurring is 1 minus the probability that it doesn't occur This rule is useful because sometimes it is easier to calculate the probability of an event happening by determining the probability of it not happening and subtracting the result from 1. Notice that the events A and are mutually exclusive and collectively exhaustive. Therefore, the probabilities of A and sum to 1. A Venn diagram illustrating the complement rule is shown as:
Example : General Rule of Multiplication A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white and the others blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not do laundry. What is the likelihood both shirts selected are white? Incidentally, it is assumed that this experiment was conducted without replacement. That is, the first shirt was not laundered and put back in the closet before the second was selected. So the outcome of the second event is conditional or dependent on the outcome of the first event.
The event that the first shirt selected is white is W1. The probability is P(W1) = 9/12 because 9 of the 12 shirts are white. The event that the second shirt selected is also white is identified as W2. The conditional probability that the second shirt selected is white, given that the first shirt selected is also white, is P(W2|W1) = 8/11. To determine the probability of 2 white shirts being selected, we use formula (5-6). P(W1 and W2) = P(W1)*P(W2 | W1) = 9/12 x 8/11 = .55 So the likelihood of selecting two shirts and finding them both to be white is .55.
Example of General Rule of Addition Suppose, for illustration, that the Florida Tourist Commission selected a sample of 200 tourists who visited the state during the year. • The survey revealed that 120 tourists went to Disney World and • 100 went to Busch Gardens near Tampa. If the special rule of addition is used, the probability of selecting a tourist who went to Disney World is .60, found by 120/200. Similarly, the probability of a tourist going to Busch Gardens is .50. The sum of these probabilities is 1.10. We know, however, that this probability cannot be greater than 1.
The explanation is that many tourists visited both attractions and are being counted twice! A check of the survey responses revealed that 60 out of 200 sampled did, in fact, visit both attractions. Not mutually exclusive. To answer our question, "What is the probability a selected person visited either Disney World or Busch Gardens?" P(A or B) = P(A) + P(B) - P(A and B) = P(Disney) + P(Busch) - P(Disney & Busch) = .60 + .50 - .30 = .80
English logician J. Venn (1834-1923) developed a diagram to portray graphically the outcome of an experiment.
The mutually exclusive concept and various other rules for combining probabilities can be illustrated using this device. To construct a Venn diagram, a space is first enclosed representing the total of all possible outcomes. This space is usually in the form of a rectangle. An event is then represented by a circular area which is drawn inside the rectangle proportional to the probability of the event. The following Venn diagram represents the mutually exclusive concept. There is no overlapping of events, meaning that the events are mutually exclusive. In the following diagram, assume the events A, B, and C are about equally likely.
Independence
The occurrence of one event has no effect on the probability of the occurrence of another event. One way to think about independence is to assume that events A and B occur at different times. For example, when event B occurs after event A occurs, does A have any effect on the likelihood that event B occurs? If the answer is no, then A and B are independent events. To illustrate independence, suppose two coins are tossed. The outcome of a coin toss (head or tail) is unaffected by the outcome of any other prior coin toss (head or tail).
Empirical Probability aka Relative Frequency
The probability of an event happening is the fraction of the time similar events happened in the past. = # of times the event occurs Total number of observations The empirical approach to probability is based on what is called the law of large numbers. The key to establishing probabilities empirically is that more observations will provide a more accurate estimate of the probability. This reasoning allows us to use the empirical or relative frequency approach to finding a probability. Here are some examples. • Last semester, 80 students registered for Business Statistics 101 at Scandia University. Twelve students earned an A. Based on this information and the empirical approach to assigning a probability, we estimate the likelihood a student will earn an A is .15. • Kobe Bryant of the Los Angeles Lakers made 403 out of 491 free throw attempts during the 2009-10 NBA season. Based on the empirical rule of probability, the likelihood of him making his next free throw attempt is .821.
Example of Complement Rule Recall the probability a bag of mixed vegetables is underweight is .025 and the probability of an overweight bag is .075. Use the complement rule to show the probability of a satisfactory bag is .900. Show the solution using a Venn diagram.
The probability the bag is unsatisfactory equals the probability the bag is overweight plus the probability it is underweight. That is, P(A or C) = P(A) + P(C) = .025 + .075 = .100. The bag is satisfactory if it is not underweight or overweight, so P(B) = 1 - [P(A) + P(C)] = 1 - [.025 + .075] = 0.900. The Venn diagram portraying this situation is:
mutually exclusive events
Two events that cannot occur at the same time The variable "gender" presents mutually exclusive outcomes, male and female. An employee selected at random is either male or female but cannot be both. A manufactured part is acceptable or unacceptable. The part cannot be both acceptable and unacceptable at the same time. In a sample of manufactured parts, the event of selecting an unacceptable part and the event of selecting an acceptable part are mutually exclusive.
subjective probability Subjective Concept of Probability
Uses a probability value based on an educated guess or estimate, employing opinions and inexact information. The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available. Illustrations of subjective probability are: 1. Estimating the likelihood the New England Patriots will play in the Super Bowl next year. 2. Estimating the likelihood you will be married before the age of 30. 3. Estimating the likelihood the U.S. budget deficit will be reduced by half in the next 10 years.
Objective Probability : Classical Classical probability is based on the assumption that the outcomes of an experiment are equally likely.
Using the classical viewpoint, the probability of an event happening is computed by dividing the number of favorable outcomes by the number of possible outcomes.
Outcome
a particular result of an experiment
collectively exhaustive
at least one of the events must occur when an experiment is conducted If an experiment has a set of events that includes every possible outcome, such as the events "an even number" and "an odd number" in the die-tossing experiment, then the set of events is collectively exhaustive. For the die-tossing experiment, every outcome will be either even or odd. So the set is collectively exhaustive.
Joint Probability The following Venn diagram shows two events that are not mutually exclusive. The two events overlap to illustrate the joint event that some people have visited both attractions.
the probability of the intersection of two events A probability that measures the likelihood two or more events will happen concurrently.
