DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics Introduction to Business Statistics QM 120 Ch Chapter 4 t 4 Spring 2008 Dr. Mohammad Zainal
Chapter 4: Experiment, outcomes, and sample space 2 2 � Probability and statistics are related in an important way. It is used to allow us to evaluate the reliability of our conclusions about the population when we have only sample information. � Data are obtained by observing either uncontrolled events in nature or controlled situation in laboratory. We use the term term experiment experiment to to describe describe either either method method of of data data collection. � The e observation obse vatio or o measurement easu e e t generated ge e ated by by an a experiment may or may not produce a numerical value. Here are some examples: � Recording a test grade � Interviewing a householder to obtain his or her opinion in certain issue issue. QMIS 120, CH 4
Chapter 4: Experiment, outcomes, and sample space 3 3 Table 1: Examples of experiments, outcomes, and sample spaces Experiment Outcomes Sample Space Toss a coin once Head, Tail S = {Head, Tail} Roll a die once 1, 2, 3, 4, 5, 6 S = {1, 2, 3, 4, 5, 6} Play a lottery Win, Lose S = {Win, Lose} Select a student M, F S = {M,F} Toss a coin twice Toss a coin twice HH, HT, TH, TT HH HT TH TT S = { HH HT TH TT } S { HH, HT, TH, TT } � Venn diagram is a picture that depicts all possible outcomes for an experiment while in tree diagram, each outcome is represented a branch of a tree. H Venn Tree ● T ● H Diagram Diagram T T QMIS 120, CH 4
Chapter 4: Experiment, outcomes, and sample space 4 4 Example: Draw the Venn and tree diagrams for the experiment of tossing a coin twice. Solution : Example: Draw a tree diagram for three tosses of a coin. List E a le D a a t ee dia a fo th ee to e of a oi Li t all outcomes for this experiment in a sample space S. Solution QMIS 120, CH 4
Chapter 4: Experiment, outcomes, and sample space 5 5 � A simple event is the outcome that is observed on a single repetition of the experiment. It is often denoted by E with a p p y subscript. Example: Toss a die and observe the number that appears on Example: Toss a die and observe the number that appears on the upper face. List the simple event in the experiment. Solution: Solution: QMIS 120, CH 4
Chapter 4: Experiment, outcomes, and sample space 6 6 � We can now define an event (or compound event) as a collection of simple events, often denoted a capital letter. Example: Tossing a die (continued) We can define the events A and B as follow, A: Observe an odd number B: Observe a number less than 4 S l ti Solution: QMIS 120, CH 4
Chapter 4: Experiment, outcomes, and sample space 7 7 Example: Suppose we randomly select two persons from members of a club and observe whether the person selected is a man or woman. Write all the outcomes from experiment. Draw the Venn and tree diagrams for this experiment. Solution: Solution: � Two events are mutually exclusive if, when one event occurs, the other cannot, and vice versa. � The set of all simple events is called the sample space, S. QMIS 120, CH 4
Chapter 4: Calculating Probability 8 8 � Probability is a numerical measure of the likelihood that an event will occur. Two properties of probability � � The probability of an even always lies in the range 0 to 1 � The probability of an even always lies in the range 0 to 1. 0 ≤ P(E i ) ≤ 1 0 ≤ P(A) ≤ 1 0 ≤ P(A) ≤ 1 � The sum of the probabilities of all simple events for an experiment, denoted by Σ P(Ei ), is always 1. y ) y ∑ P(E i ) = P(E 1 ) + P(E 2 ) + P(E 3 ) + . . . . = 1 QMIS 120, CH 4
Chapter 4: Calculating Probability 9 9 Three conceptual approaches to probability 1 Classical probability 1. Classical probability � Two or more events that have the same probability of occurrence are said to be equally likely events. � The probability of a simple event is equal to 1 divided by the total number of all final outcomes for an equally likely experiment. � Classical probability rule to find probability 1 = = ( ( ) ) P E P E i Total number of outcomes Number of outcomes favorable to A = ( ) P A Total number of outcomes QMIS 120, CH 4
Chapter 4: Calculating Probability 10 10 Example: Find the probability of obtaining a head and the probability of obtaining a tail for one toss of a coin probability of obtaining a tail for one toss of a coin. Solution QMIS 120, CH 4
Chapter 4: Calculating Probability 11 11 Example: Find the probability of obtaining an even number in one roll of a die. Solution QMIS 120, CH 4
Chapter 4: Calculating Probability 12 12 � Calculating the probability of an event: � List all the simple events in the sample space. � List all the simple events in the sample space. � Assign an appropriate probability to each simple event. � Determine which simple events result in the event of interest. � Determine which simple events result in the event of interest. � Sum the probabilities of the simple events that result in the event of interest. Always 1. Include all simple events in the sample space. 2. Assign realistic probabilities to the simple events. QMIS 120, CH 4
Chapter 4: Calculating Probability 13 13 Example: A six years boy has a safe box that contains four banknotes: One ‐ Dinar, Five ‐ Dinar, Ten ‐ Dinar, Twenty ‐ Dinar. His sister which is a three years old girl randomly grabbed three banknotes from the safe box to buy a 30 KD toy. Find the odds (probability) that this girl can buy the toy odds (probability) that this girl can buy the toy. Solution QMIS 120, CH 4
Chapter 4: Calculating Probability 14 14 2. Relative frequency concept of probability � Suppose we want to know the following probabilities: � Suppose we want to know the following probabilities: � The next car coming out of an auto factory is a “lemon” � A randomly selected family owns a home � A randomly selected woman has never smoked � The outcomes above are neither equally likely nor fixed for each sample. l � The variation goes to zero as n becomes larger and larger � If an experiment is repeated n times and an event A is observed f If i i d i d A i b d f times, then f = ( ) P A n QMIS 120, CH 4
Chapter 4: Calculating Probability 15 15 Example: In a group of 500 women, 80 have played golf at least once. Suppose one of these 500 woman is selected. What is the probability that she played golf at least once Solution QMIS 120, CH 4
Calculating Probability 16 16 Example: Ten of the 500 randomly selected cars manufactured at a certain auto factory are found to be lemons. What is the y probability that the next car manufactured at that factory is a lemon? S l Solution: i QMIS 120, CH 4
Calculating Probability 17 17 Example: Lucca Tool Rental would like to assign probabilities to the number of car polishers it rents each day. Office records p y show the following frequencies of daily rentals for the last 40 days. Number of Polishers Rented Number of Days 0 4 1 6 2 2 18 18 3 10 4 2 QMIS 120, CH 4
Calculating Probability 18 18 � Law of large numbers: If an experiment is repeated again and again, the probability of an event obtained from the relative frequency approaches the actual probability. 3. Subjective probability � Suppose we want to know the following probabilities: pp g p � A student who is taking a statistics class will get an A grade. � KSE price index will be higher at the end of the day. � China will dominate the gold medal list in the 2008 Olympics. � Subjective probability is the probability assigned to an event based on subjective judgment experience information and belief subjective judgment, experience, information, and belief. QMIS 120, CH 4
Counting Rule 19 19 � Suppose that an experiment involves a large number N of simple events and you know that all the simple events are equally likely. Then each simple event has probability 1/N and the probability of an event A can be calculated as n = A ( ) P A N Where is the number of simple events that result in event A n A � The mn rule � Consider an experiment that is performed in two stages. If the first stage can be performed in m ways and for each of these ways, the second stage can be accomplished in n ways, then there mn ways to accomplish the experiment QMIS 120, CH 4
Counting Rule 20 20 Example: Suppose you want to order a car in one of three styles and in one of four paint colors. To find out how many options are available you can options are available, you can think of first picking one of the m = 3 styles and then one of n = 4 colors. QMIS 120, CH 4
Counting Rule 21 21 � The extended mn rule � If an experiment is performed in k stages, with n 1 ways to accomplish � If an experiment is performed in k stages, with n 1 ways to accomplish the first stage, n 2 to accomplish the second stage…, and n k ways to accomplish the k th stage, the number of ways to accomplish the experiment is p 3 ... n n n n 1 2 k Example: A bus driver can take three routes from city A to city B, four routes from city B to city C, and three routes from city C t C to city D. For traveling from A to D, the driver must drive it D F t li f A t D th d i t d i from A to B to C to D, how many possible routes from A to D are available are available QMIS 120, CH 4
Recommend
More recommend
