Probability is the likelihood that an event will occur.

Transcription

1 Section 3.1 Basic Concepts of is the likelihood that an event will occur. In Chapters 3 and 4, we will discuss basic concepts of probability and find the probability of a given event occurring. Our main purpose for studying basic concepts of probability is to help in our understanding of statistical procedures that use a probability as the basis for making a decision. In Section 3.1, we introduce basic definitions and properties of probability. In Section 3.2, we discuss conditional probability and the probability of two events occurring in sequence. In Section 3.3, we will find the probability of at least one of two events occurring.

2 Important Definitions Experiment A probability experiment is any action through which a specific result is obtained. The action may be a physical action, such as tossing a coin or rolling a die, or a simulation, such as generating random integers with a calculator or computer, or simply an observation, such as recording the color of a vehicle that sits in a parking lot The specific results could be a measurement, a count, a response or an observation. Outcome The result of a single trial of a probability experiment is called an outcome.

3 Sample Space The set of all possible outcomes of a probability experiment is called the sample space of the probability experiment. Event An event is a subset of the sample space and consists of one or more outcomes. Events are typically denoted with a capital letter.

4 Example 1 A standard six-sided die is rolled. Identify: (a) the probability experiment (b) the sample space of the probability experiment (c) the event E = the face showing is an odd number (d) the outcome that the face showing is 6 (a) (b) (c) (d) The probability experiment is the action itself. Thus, rolling a standard six-sided die is the probability experiment. The sample space is set of all possible outcomes of the probability experiment. Thus, the sample space is {1,2,3,4,5,6}. The event E = the face showing is an odd number consists of all members of the sample space that are odd numbers. Thus, this event is defined as E = {1,3,5}. An outcome is the result of a single trial of a probability experiment. The outcome that the face showing is 6 is {6}.

5 Example 2 A prime number less than 10 is randomly selected. Identify: (a) the probability experiment (b) the sample space of the probability experiment (c) the event E = the number selected is odd (d) the outcome that the number selected is 2 (a) (b) (c) (d) The probability experiment is the action itself. Thus, randomly selecting a prime number less than 10 is the probability experiment. The sample space is set of all possible outcomes of the probability experiment. Thus, the sample space is {2,3,5,7}. The event E = number selected is odd is the event described. Thus, this event is defined as E = {3,5,7}. An outcome is the result of a single trial of a probability experiment. The outcome that the number selected is 2 is {2}.

6 Example 3 A card is randomly selected from a well-shuffled deck of standard playing cards. Identify: (a) (b) (c) (d) the probability experiment the sample space of the probability experiment the event E = the card selected is a king the outcome that the number selected is ace of spades (a) (b) The probability experiment is the action itself. Thus, randomly selecting a card from a well-shuffled deck of standard playing cards is the probability experiment. The sample space is set of all possible outcomes of the probability experiment. Thus, the sample space has 52 members:

7 (c) (d) The event E = the card selected is a king is the event described. Thus, this event is defined as {KH, KD, KS, KC}, where K = king, H = hearts, D = diamonds, S = spades and C = clubs. An outcome is the result of a single trial of a probability experiment. The outcome that the card selected is the ace of spades is {AS}, where A = ace and S = spades. The events described in part (c) of each of these examples consisted of more than one outcome. If we change part (c) of Example 3 to the card selected is the king of hearts, then the event would consist of only one outcome. An event in which there is only one outcome is called a simple event.

8 Example 4 Decide whether the event described is a simple event or not. (a) You ask one of your teachers where they went to college. The event is that they answer Lycoming College. (b) You roll a standard six-sided die. The event A = the face showing is an even number. (c) You generate a random integer between 0 and 100, inclusive, on your calculator. The event B = the integer generated is a multiple of 5. (d) You randomly select one card from a well-shuffled deck of standard playing cards. The event C = the card selected is the 10 of diamonds. (e) You randomly select a student in this class. The event M = the student s first name is Maggie.

9 (a) You ask one of your teachers where they went to college. The event is that they answer Lycoming College. Since there is only one outcome in this event, this is a simple event. (b) You roll a standard six-sided die. The event A = the face showing is an even number. Since A = {2,4,6}, this event contains more than one outcome is not a simple event. (c) You generate a random integer between 0 and 100, inclusive, on your calculator. The event B = the integer generated is a multiple of 5. Since B = {0,5,10,15,, 85, 90, 95, 100}, this event contains more than one outcome is not a simple event.

10 (d) You randomly select one card from a well-shuffled deck of standard playing cards. The event C = the card selected is the 10 of diamonds. Since C = {10 of diamonds}, there is only one outcome in this event so this is a simple event. (e) You randomly select a student in this class. The event M = the student s first name is Maggie. Since M = {Maggie No.1, Maggie No.2, Maggie No.3}, there are three outcomes in this event. This is not a simple event.

11 Range of Probabilities measures the likelihood of event occurring and is expressed numerically as a fraction, decimal or percent. We will use the notation P(E) to denote the probability of event occurring. If an event E is an impossible event one that will definitely not occur then P(E) = 0. E = Mr. Smith can dunk a basketball on a regulation height backboard without the aid of a ladder, trampoline or other height-assisting device. Since E is an impossible event, P(E) = 0.

12 If an event E is a certain event one that will definitely occur then P(E) = 1. E = Students in 7 th grade will take the PSSA this school year. Since E is a certain event, P(E) = 1. If an event E is equally like to occur as it is to not occur, then P(E) = 0.5. E = A fair coin tossed and it lands with heads showing. Since E is as likely to occur as it is not occur, then P(E) = 0.5. Range of Probabilities: 0 P E 1 An event in which P(E) 0.05 is considered to be an unusual event.

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14 Classical or Theoretical If a sample space can be determined for a probability experiment, and the number of outcomes in a certain event can be determined and all events are equally like to occur, then the probability of an event E occurring can be computed using the classical or theoretical probability formula: P E number of outcomes in event E total number of outcomes in the sample space Example 5 Experiment = Tossing a coin Sample space = {Heads,Tails} Event A = Tossing heads A = {Heads} P A number of outcomes in event A 1 total number of outcomes in sample space 2

15 Example 6 Experiment = Selecting a card at random from a well-shuffled deck of standard playing cards Sample space {HA,HK,HQ,HJ,H10,H9,H8,H7,H6,H5,H4,H3,H2,DA,DK,DQ,DJ,D10,D9,D8,D7,D6,D5,D4, D3,D2,CA,CK,CQ,CJ,C10,C9,C8,C7,C6,C5,C4,C3,C2,SA,SK,SQ,SJ,S10,S9,S8,S7,S6,S5,S4, S3,S2} Event A = Selecting a king Event B = Selecting a face card Event C = Selecting a red 10 A = {HK,DK,CK,SK} B = {HK,HQ,HJ,DK,DQ,DJ,CK,CQ,CJ,SK,SQ,SJ} C = {H10,D10} P A P B P C number of outcomes in event A 4 1 total number of outcomes in sample space number of outcomes in event B 12 3 total number of outcomes in sample space number of outcomes in event C 2 1 total number of outcomes in sample space 52 26

16 Empirical or Statistical If an experiment is repeated over and over, regular patterns are often formed in the results. In such a case, the probability of an event E occurring can be computed using the empirical or statistical probability formula: P E frequency of event E total frequency The empirical probability formula can be used even if each outcome in the sample space is not equally likely to occur. Example 7 Experiment = Selecting a candy at random from a bag of Reese s Pieces Event O = selecting an orange candy Number of orange candies = 26 Number of yellow candies = 18 Number of brown candies = 13 frequency of event O PO total frequency Round probabilities to three (3) decimals.

17 Example 8 Consider the ages of 1,000 employees at a company shown in the table below. Ages Frequency to to to to One employee is selected at random. Find each probability. (a) What is the probability that the employee selected is between 25 and 34 years old? frequency between 25 and P25 to total frequency 1, (b) What is the probability that the employee selected is between 45 and 54 years old? frequency between 45 and P45 to total frequency 1, (c) What is the probability that the employee selected is at least 25 years old? frequency 25 or older Pat least total frequency 1, 000 1,

18 In the last question of Example 8, we could have used complementary events to determine the probability that an employee selected was at least 25. The complement of event E, denoted E and read E prime, is defined as the set of all outcomes in the sample space of a probability experiment that are not included in event E. The probability of an event and the probability of the complement of the event must have a sum of 1. P E P E 1 P E 1 P E P E 1 P E (c) What is the probability that the employee selected is at least 25 years old? The complement of this event is all employees that are less than 25 years old. Since 15 to 24 is the first class given in the table and frequency of this class is 54, we could have computed the probability in part (c) as follows: E = employee selected is at least 25 years old E = employee selected is younger than 25 years old PE 1 PE , 000 1,

19 Subjective When the chance of precipitation is given by a meteorologist, it is typically given as a percent. While the percent is based on computer models, how does the meteorologist distinguish between a 10% chance and a 20% chance of precipitation? The probability of precipitation is an example of a subjective probability, one that is based upon intuition, estimates, educated guesses or simply a person s measure of belief that a specified event will occur. Other statements that illustrate subjective probability are: I believe there is a 75% probability that the Dow Jones Industrial Average will drop 3% tomorrow. The probability that the Miami Heat repeat as NBA champions this year is 90%. The probability of landing a man on Mars by the year 2020 is 25%.

20 The Law of Large Numbers If a probability of an event is unknown such as the probability of selecting an orange candy when randomly selecting one candy from a bag of Reese s Pieces experimentation may be used to approximate the probability. (Hershey Company keeps the proportion of each color of candies in a bag of Reese s Pieces as a closely guarded secret.) The results of extensive sampling of 1.53-ounce bags of Reese s Pieces candies is shown in the table below. Color of Candy Frequency Orange 22,331 Yellow 12,974 Brown 17,435 Total 52,740

21 Color of Candy Frequency Orange 22,331 Yellow 12,974 Brown 17,435 Total 52,740 Example 9 If a candy is selected at random from a bag of Reese s Pieces, what is the probability that the candy is: (a) orange? (b) yellow? (c) brown? (a) number of orange candies 22,331 Porange total number of candies 52,740 (b) number of yellow candies 12,974 Pyellow total number of candies 52,740 (c) number of brown candies 12,974 Pbrown total number of candies 52,740

22 While it is uncertain whether these are the actual probabilities, they are likely close to the actual values. This is due to The Law of Large Numbers. The Law of Large Numbers If a probability experiment is repeated over and over, then the empirical probability of an event approaches the theoretical probability of the event. As the number of repetitions/trials increases, the empirical probability becomes even closer to the actual probability of the event occurring.

23 Relative Frequency Relative Frequency The graph below shows the results of rolling a standard sixsided die 25 times. The graph below shows the results of rolling a standard sixsided die 175 times. Rolling a Standard Six-Sided Die Rolling a Standard Six-Sided Die Face Showing Face Showing As the number of trials increased, the relative frequency bar graph began to approach the theoretical distribution of this experiment a uniform distribution in which the relative frequency of each outcome is 1/6.