PROBABILITY section. The Probability of an Event

Size: px
Start display at page:

Download "PROBABILITY 14.3. section. The Probability of an Event"

Transcription

1 4.3 Probability (4-3) PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques to find probabilities. The Probability of an Event The Addition Rule Complementary Events Odds The Probability of an Event In probability an experiment is a process such as tossing a coin, tossing a die, drawing a poker hand from a deck, or arranging people in a line. A sample space is the set of all possible outcomes to an experiment. An event is a subset of a sample space. For example, if we toss a coin, then the sample space consists of two equally likely outcomes, heads and tails. We write S H, T. The subset E H is the event of getting heads when the coin is tossed. We use n(s) to represent the number of equally likely outcomes in the sample space S and n(e ) to represent the number of outcomes in the event E. For the example of tossing a coin, n(s) 2 and n(e). The Probability of an Event If S is a sample space of equally likely outcomes to an experiment and the event E is a subset of S, then the probability of E, P(E), is defined to be n(e) P(E). n(s) When S H, T and E H, n(e) P(E) 2. n(s) So the probability of getting heads on a single toss of a coin is. 2 If E is the event of getting 2 heads on a single toss of a coin, then n(e) 0 and P(E) If E is the event of getting fewer than 2 heads on a single toss of a coin, then for either outcome H or T we have fewer than 2 heads. So E H, T, n(e) 2, and P(E) 2 2. Note that the probability of an event is a number between 0 and inclusive, being the probability of an event that is certain to occur and 0 being the probability of an event that is impossible to occur. E X A M P L E Rolling a die What is the probability of getting a number larger than 4 when a single die is rolled? When we roll a die, we count the number of dots showing on the upper face of the die. So the sample space of equally likely outcomes is S, 2, 3, 4, 5, 6. Since only 5 and 6 are larger than 4, E 5, 6. According to the definition of probability, n(e) P(E) n(s)

2 728 (4-4) Chapter 4 Counting and Probability M A T H A T W O R K The probability experiments discussed in this chapter are not just textbook examples that have no relationship to real life. For example, if a couple plans to have 6 children and the probability of having a girl on each try is 2, then the couple can expect to have 3 girls. If you guess at the answer to each question of a 00-question, 5-choice multiple-choice test, then you have 5 probability of getting each question correct, and you can expect to get 20 questions correct. Try it. The expected number of successes is the product of the probability of success and the number of tries. Lotteries provide us an opportunity to observe massive probability experiments. In the Florida Lottery you can win by choosing 6 numbers from the numbers through 49 and matching the 6 numbers chosen by the Florida Lottery. There are C(49, 6) ways to choose 6 numbers from 49, so the probability of winning on any individual try is. C(49, 6) 3,98 3,86 In the fall of 990 the weekly drawing frequently had relatively few participants, and consequently there was no winner for many weeks. When the prize got up to $06.5 million, the lottery got national attention. People came from everywhere to participate. During the week prior to September 5, 990, 09,63,978 tickets were sold. We expected 3,98 09,63, winners. On September 5 the 3,86 winning numbers were announced, and 6 winners shared the prize. Of course, probability cannot predict the future like a fortune-teller, but the power of probability to tell us what to expect is truly amazing. LOTTERIES E X A M P L E 2 Tossing coins What is the probability of getting at least one head when a pair of coins is tossed? Since there are 2 equally likely outcomes for the first coin and 2 equally likely outcomes for the second coin, by the fundamental counting principle there are 4 equally likely outcomes to the experiment of tossing a pair of coins. We can list the outcomes as ordered pairs: S (H, H), (H, T), (T, H), (T, T). Since 3 of these outcomes result in at least one head, E (H, H), (H, T), (T, H), and n(e) 3. So n(e) P(E) 3 4. n(s) E X A M P L E 3 Rolling a pair of dice What is the probability of getting a sum of 6 when a pair of dice is rolled?

3 Since there are 6 equally likely outcomes for each die, there are equally likely outcomes to the experiment of rolling the pair. We can list the outcomes as ordered pairs: S (, ), (, 2), (, 3), (, 4), (, 5), (, 6), (2, ), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, ), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, ), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, ), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, ), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) 4.3 Probability (4-5) 729 The sum of the numbers is 6, describes the event E (5, ), (4, 2), (3, 3), (2, 4), (, 5). So n(e) 5 P(E). n(s) The Addition Rule In tossing a pair of dice, let A be the event that doubles occurs and B be the event that the sum is 4. We can list the following events and their probabilities: 6 A (, ), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) and P(A) 3 B (3, ), (2, 2), (, 3) and P(B) A B (, ), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (3, ), (, 3) and P(A B) 3 86 A B (2, 2) and P(A B) 8 Note that the probability of doubles or a sum of 4, P(A B), is and This equation makes sense because there is one outcome, (2, 2), that is in both the events A and B. This example illustrates the addition rule. The Addition Rule If A and B are any events in a sample space, then P(A B) P(A) P(B) P(A B). If P(A B) 0, then A and B are called mutually exclusive events and P(A B) P(A) P(B). Note that for mutually exclusive events it is impossible for both events to occur. The addition rule for mutually exclusive events is a special case of the general addition rule.

4 730 (4-6) Chapter 4 Counting and Probability E X A M P L E 4 E X A M P L E 5 The addition rule At Downtown College 60% of the students are commuters (C), 50% are female (F), and 30% are female commuters. If a student is selected at random, what is the probability that the student is either a female or a commuter? By the addition rule the probability of selecting either a female or a commuter is P(F C) P(F) P(C) P(F C) The addition rule with dice In rolling a pair of dice, what is the probability that the sum is 2 or at least one die shows a 2? Let A be the event that the sum is 2 and B be the event that at least one die shows a 2. Since A occurs on only one of the equally likely outcomes (see Example 3), P(A). Since B occurs on of the equally likely outcomes, P(B). Since A and B are mutually exclusive, we have P(A B) P(A) P(B) 2 3. Complementary Events If the probability of rain today is 60%, then the probability that it does not rain is 40%. Rain and not rain are called complementary events. There is no possibility that both occur, and one of them must occur. If A is an event, then A (read A bar or A complement ) represents the complement of the event A. Note that complementary events are mutually exclusive, but mutually exclusive events are not necessarily complementary. Complementary Events Two events A and A are called complementary events if A A and P(A) P(A). E X A M P L E 6 E X A M P L E 7 Complementary events What is the probability of getting a number less than or equal to 4 when rolling a single die? We saw in Example that getting a number larger than 4 when rolling a single die has probability 3. The complement to getting a number larger than 4 is getting a number less than or equal to 4. So the probability of getting a number less than or equal to 4 is 2 3. Complementary events If the probability that White Lightning will win the Kentucky Derby is 0.5, then what is the probability that White Lightning does not win the Kentucky Derby?

5 4.3 Probability (4-7) 73 study tip Study for the final exam by working actual test questions. Be sure to rework all of your tests. Do the chapter tests in this book. You can get more tests to work by asking students or instructors for tests that were given in other classes of this course. Let W be winning the Kentucky Derby and N be not winning the Kentucky Derby. Since W and N are complementary events, we have P(W) P(N). So P(N) P(W) Odds If the probability is 2 3 that the Giants win the Super Bowl and 3 that they lose, then they are twice as likely to win as they are to lose. We say that the odds in favor of the Giants winning the Super Bowl are 2 to. Notice that odds are not probabilities. Odds are ratios of probabilities. We usually write odds as ratios of whole numbers. Odds If A is any event, then the odds in favor of A is the ratio P(A) to P(A) and the odds against A is the ratio of P(A) to P(A). E X A M P L E 8 Determining odds What are the odds in favor of getting a sum of 6 when rolling a pair of dice? What are the odds against a sum of 6? 5 In Example 3 we found the probability of a sum of 6 to be. So the probability of the complement (the sum is not 6) is 3. The odds in favor of getting a sum of 6 are 5 to 3. Multiply each fraction by to get the odds 5 to 3. The odds against a sum of 6 are 3 to 5. E X A M P L E 9 helpful hint Odds and probability are often confused, even by people who write lottery tickets. If the probability of winning a lottery is, then the probability of losing is, and the 00 odds in favor of winning are to 99. Many lottery tickets will state (incorrectly) that the odds in favor of winning are to 00. Determining probability given the odds If the odds in favor of Daddy s Darling winning the third race at Delta Downs are 4 to, then what is the probability that Daddy s Darling wins the third race? Since 4 to is the ratio of the probability of winning to not winning, the probability of winning is four times as large as the probability of not winning. Let P(W) x and P(W ) 4x. Since P(W ) P(W ), we have 4x x, or 5x, or x 5. So the probability of winning is 4 5. We can write the idea found in Example 9 as a strategy for converting from odds to probabilities. Strategy for Converting from Odds to Probability If the odds in favor of event E are a to b, then a b P(E) and P(E). a b a b

6 732 (4-8) Chapter 4 Counting and Probability WARM-UPS True or false? Explain your answer.. If S is a sample space of equally likely outcomes and E is a subset of S, then P(E) n(e). False 2. If an experiment consists of tossing 3 coins, then the sample space consists of 6 equally likely outcomes. False 3. The probability of getting at least one tail when a coin is tossed twice is True 4. The probability of getting at least one 4 when a pair of dice is tossed is. True 5. The probability of getting at least one head when 5 coins are tossed is False 6. If 3 coins are tossed, then getting exactly 3 heads and getting exactly 3 tails are complementary events. False 7. If the probability of getting exactly 3 tails in a toss of 3 coins is 8, then the probability of getting at least one head is 7 8. True 8. If the probability of snow today is 80%, then the odds in favor of snow are 8 to 0. False 9. If the odds in favor of an event E are 2 to 3, then P(E) 2 3. False 0. The ratio of 2 to 3 is equivalent to the ratio of 2 to 3. False 4. 3 EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is an experiment? An experiment is a process for which the outcomes are uncertain. 2. What is a sample space? A sample space is the set of all possible outcomes to an experiment. 3. What is an event? An event is a subset of a sample space. 4. What is the probability of an event? The probability of an event is the ratio of the number of outcomes in the event to the number of outcomes in the sample space. 5. What is the addition rule? The addition rule states that if A and B are events in a sample space, then P(A B) P(A) P(B) P(A B). 6. What are the odds in favor of an event? The odds in favor of an event is the ratio of the probability of the event to the probability of the complement of the event. Solve each probability problem. See Example If a single die is tossed, then what is the probability of getting a) a number larger than 3? b) a number less than or equal to 5? c) a number other than 6? d) a number larger than 7? e) a number smaller than 9? 2, 5 6, 5 6,0, 8. If a single coin is tossed once, then what is the probability of getting a) tails? c) exactly three heads? b) fewer than two heads? 2,,0 9. If a pair of coins is tossed, then what is the probability of getting a) exactly two heads? c) exactly two tails? b) at least one tail? d) at most one tail? 4, 3 4, 4, 3 4

7 4.3 Probability (4-9) If a single coin is tossed twice, then what is the probability of getting a) heads followed by tails? c) a tail on the second toss? b) two heads in a row? d) exactly one tail? 4, 4, 2, 2. If a pair of dice is tossed, then what is the probability of getting a) a pair of 2 s? d) a sum greater than? b) at least one 2? e) a sum less than 2? c) a sum of 7?, 6,,,0 2. If a single die is tossed twice, then what is the probability of getting a) a followed by a 2? b) a sum of 3? 8 c) a 6 on the second toss? 6 d) no more than two 5 s? e) an even number followed by an odd number? 4 3. A ball is selected at random from a jar containing 3 red balls, 4 yellow balls, and 5 green balls. What is the probability that a) the ball is red? 4 b) the ball is not yellow? 2 3 c) the ball is either red or green? 2 3 d) the ball is neither red nor green? 3 FIGURE FOR EXERCISE 3 4. A committee consists of Democrat, 5 Republicans, and 6 independents. If one person is randomly selected from the committee to be the chairperson, then what is the probability that a) the person is a Democrat? 2 b) the person is either a Democrat or a Republican? 2 7 c) the person is not a Republican? 2 5. A jar contains 0 balls numbered through 0. Two balls are randomly selected one at a time without replacement. What is the probability that a) is selected first and 2 is selected second? 9 0 b) the sum of the numbers selected is 3? c) the sum of the numbers selected is 6? A small company consists of a president, a vice-president, and 4 salespeople. If 2 of the 6 people are randomly selected to win a Hawaiian vacation, then what is the probability that none of the salespeople is a winner? If a 5-card poker hand is drawn from a deck of 52, then what is the probability that a) the hand contains the ace, king, queen, jack, and ten of spades? 2,59 8,960 b) the hand contains one 2, one 3, one 4, one 5, and one 6? 024 2,598, If 5 people with different names and different weights randomly line up to buy concert tickets, then what is the probability that a) they line up in alphabetical order? 20 b) they line up in order of increasing weight? 20 Use the addition rule to solve each problem. See Examples 4 and Among the drivers insured by American Insurance, 65% are women, 38% of the drivers are in a high-risk category, and 24% of the drivers are high-risk women. If a driver is randomly selected from that company, what is the probability that the driver is either high-risk or a woman? What is the probability of getting either a sum of 7 or at 5 least one 4 in the toss of a pair of dice? 2 2. A couple plans to have 3 children. Assuming males and females are equally likely, what is the probability that they have either 3 boys or 3 girls What is the probability of getting a sum of 0 or a sum of 5 7 in the toss of a pair of dice? 23. What is the probability of getting either a heart or an ace 4 when drawing a single card from a deck of 52 cards? What is the probability of getting either a heart or a spade when drawing a single card from a deck of 52 cards? 2 Solve each problem. See Examples 6 and If the probability of surviving a head-on car accident at 55 mph is 0.005, then what is the probability of not surviving? If the probability of a tax return not being audited by the IRS is 0.97, then what is the probability of a tax return being audited? A pair of dice is tossed. What is the probability of a) getting a pair of 4 s? b) not getting a pair of 4 s? 3 5 c) getting at least one number that is not a 4? Three coins are tossed. What is the probability of a) getting three heads? 8 b) not getting three heads? 7 8 c) getting at least one tail? 7 8

8 734 (4-20) Chapter 4 Counting and Probability Solve each problem. See Examples 8 and If the probability is 60% that the eye of Hurricane Edna comes ashore within 30 miles of Charleston, then what are the odds in favor of the eye of Edna coming ashore within 30 miles of Charleston? 3 to 2 FIGURE FOR EXERCISE 38 FIGURE FOR EXERCISE If the probability that a Sidewinder missile hits its target is 8 9, then what are the odds a) in favor of the Sidewinder hitting its target? 8 to b) against the Sidewinder hitting its target? to 8 3. If the probability that the stock market goes up tomorrow is 3 5, then what are the odds a) in favor of the stock market going up tomorrow? 3 to 2 b) against the stock market going up tomorrow? 2 to If the probability of a coal miners strike this year is 0, then what are the odds a) in favor of a strike? 9 to b) against a strike? to If the odds are 3 to in favor of the Black Hawks winning their next game, then a) what are the odds against the Black Hawks winning their next game? to 3 b) what is the probability that the Black Hawks win their next game? If the odds are 5 to against the Democratic presidential nominee winning the election, then a) what are the odds in favor of the Democrat winning the election? to 5 b) what is the probability that the Democrat wins the election? What are the odds in favor of getting exactly 2 heads in 3 tosses of a coin? 3 to 5. What are the odds in favor of getting a 6 in a single toss of a die? to What are the odds in favor of getting a sum of 8 when tossing a pair of dice? 5 to What are the odds in favor of getting at least one 6 when tossing a pair of dice? to If one million lottery tickets are sold and only one of them is the winning ticket, then what are the odds in favor of winning if you hold a single ticket? to 999, What are the odds in favor of winning a lottery where you must choose 6 numbers from the numbers through 49? to 3,983,85 4. If the odds in favor of getting 5 heads in 5 tosses of a coin are to 3, then what is the probability of getting 5 heads in 5 tosses of a coin? If the odds against Smith winning the election are 2 to 5, then what is the probability that Smith wins the election? 5 7 GETTING MORE INVOLVED 43. In the Louisiana Lottery a player chooses 6 numbers from the numbers through 44. You win the big prize if the 6 chosen numbers match the 6 winning numbers chosen on Saturday night. a) What is the probability that you choose all 6 winning numbers? b) What is the probability that you do not get all 6 winning numbers? c) What are the odds in favor of winning the big prize with a single entry?, 7, 059, 05, to 7,059,05 7,059,052 7, 059, In the Louisiana Power Ball a player chooses 5 numbers from the numbers through 49 and one number (the power ball) from through 42. a) How many ways are there to choose the 5 numbers and, choose the power ball? b) What is the probability of winning the big prize in the Power Ball Lottery? c) What are the odds in favor of winning the big prize? 80,089,28,, to 80,089,27 80,08 9,28

1.3 Sample Spaces and Events

1.3 Sample Spaces and Events 1.3 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) 2 5

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) 2 5 MTH 64 Practice Exam 4 - Probability Theory Spring 2008 Dr. Garcia-Puente Name Section MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Give the probability

More information

Probability. Experiment - any happening for which the result is uncertain. Outcome the possible result of the experiment

Probability. Experiment - any happening for which the result is uncertain. Outcome the possible result of the experiment Probability Definitions: Experiment - any happening for which the result is uncertain Outcome the possible result of the experiment Sample space the set of all possible outcomes of the experiment Event

More information

Probability Worksheet

Probability Worksheet Probability Worksheet 1. A single die is rolled. Find the probability of rolling a 2 or an odd number. 2. Suppose that 37.4% of all college football teams had winning records in 1998, and another 24.8%

More information

Odds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes.

Odds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes. MATH 11008: Odds and Expected Value Odds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes. Suppose all outcomes in a sample space are equally likely where a of them

More information

MATH 1473: Mathematics for Critical Thinking. Study Guide and Notes for Chapter 3

MATH 1473: Mathematics for Critical Thinking. Study Guide and Notes for Chapter 3 MATH 1473: Mathematics for Critical Thinking The University of Oklahoma, Dept. of Mathematics Study Guide and Notes for Chapter 3 Compiled by John Paul Cook For use in conjunction with the course textbook:

More information

Chapter 1: Sets and Probability

Chapter 1: Sets and Probability Chapter 1: Sets and Probability Section 1.3-1.5 Sample Spaces and Events An is an activity that has observable results. An is the result of an experiment. Example 1 Examples of experiments: Flipping a

More information

Chapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.

Chapter 15. Definitions: experiment: is the act of making an observation or taking a measurement. MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.

More information

Contemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4?

Contemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4? Contemporary Mathematics- MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than

More information

CHAPTER - 16 PROBABILITY

CHAPTER - 16 PROBABILITY CHAPTER - 16 PROBABILITY 1. A box contains 25 tickets numbered from 1 to 25. One ticket is drawn at random. Find the probability of getting a number which is divisible by (i) 5 (ii) 8. 2. A die is thrown,

More information

Probability Review. 4. Six people are asked to pick a number from 1 to 20. What is the probability that at least two people will pick the same number?

Probability Review. 4. Six people are asked to pick a number from 1 to 20. What is the probability that at least two people will pick the same number? Probability Review. If two dice are rolled and their faces noted, find each of the following probabilities: the sum of the two dice is 8 b) both dice show even numbers c) the sum of the two dice is 8 OR

More information

The study of probability has increased in popularity over the years because of its wide range of practical applications.

The study of probability has increased in popularity over the years because of its wide range of practical applications. 6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,

More information

33 Probability: Some Basic Terms

33 Probability: Some Basic Terms 33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been

More information

Example: If we roll a dice and flip a coin, how many outcomes are possible?

Example: If we roll a dice and flip a coin, how many outcomes are possible? 12.5 Tree Diagrams Sample space- Sample point- Counting principle- Example: If we roll a dice and flip a coin, how many outcomes are possible? TREE DIAGRAM EXAMPLE: Use a tree diagram to show all the possible

More information

3.1 Basic Concepts of Probability and Counting

3.1 Basic Concepts of Probability and Counting EXTRA CREDIT (worth 5 exam points): Find and read an article from a newspaper or a magazine related to any topic that we have covered in this class (for example, it can involve a graph, chart, analysis,

More information

Elementary Statistics and Inference. Elementary Statistics and Inference. 14. More About Chance. 22S:025 or 7P:025. Lecture 18.

Elementary Statistics and Inference. Elementary Statistics and Inference. 14. More About Chance. 22S:025 or 7P:025. Lecture 18. Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 18 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 14 2 14. More About Chance A. Listing All Possibilities Suppose two dice

More information

In this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events.

In this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events. Lecture#4 Chapter 4: Probability In this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events. 4-2 Fundamentals Definitions:

More information

An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event

An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the

More information

Chapter 7 Probability

Chapter 7 Probability Chapter 7 Probability Section 7.1 Experiments, Sample Spaces, and Events Terminology Experiment An experiment is an activity with observable results. The results of an experiment are called outcomes of

More information

Distributions. and Probability. Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment

Distributions. and Probability. Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment C Probability and Probability Distributions APPENDIX C.1 Probability A1 C.1 Probability Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment When assigning

More information

3) Personal (or subjective) probability - personal probability is the degree of belief that an outcome will occur, based on the available information

3) Personal (or subjective) probability - personal probability is the degree of belief that an outcome will occur, based on the available information Ch. 14 Introducing Probability Def n: An experiment is a process that, when performed, results in one and only one of many observations (or outcomes). Probability is a numerical measure of likelihood that

More information

Probability Sample Test

Probability Sample Test Name: Class: Date: ID: A Probability Sample Test Multiple Choice Identify the choice that best completes the statement or answers the question.. A coin is tossed three times. What is the probability of

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Practice Test Chapter 9 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the odds. ) Two dice are rolled. What are the odds against a sum

More information

Math M118 Class Notes For Chapter 4 SECTION 4.1 The probability or (likelihood) of the occurrence of an event is found in two methods:

Math M118 Class Notes For Chapter 4 SECTION 4.1 The probability or (likelihood) of the occurrence of an event is found in two methods: Math M118 Class Notes For Chapter 4 SECTION 4.1 The probability or (likelihood) of the occurrence of an event is found in two methods: (I) Relative Frequency Method: Ex: In a survey of 100 people, it was

More information

When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to four decimal places.

When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to four decimal places. Chapter 13: General Rules of Probability Notation for Probability Event: an outcome that is usually denoted by a capital letter; A, B, C. A, B, C: Specific events P: A probability P(A): The probability

More information

Math 166 Ch 1 Sets and Probability Texas A&M Spring 2016

Math 166 Ch 1 Sets and Probability Texas A&M Spring 2016 1.1 Introduction to Sets Sets Set is a collection of items, referred to as its elements or members. A set is represented by a capital letter. For example, A = {1, 2, 3, 4, 5} is a set A containing elements

More information

CONDITIONAL PROBABILITIES

CONDITIONAL PROBABILITIES INTRODUCTION The chance of something happening gives the percentage of time it is expected to happen, when the basic process is done over and over again, independently and under the same conditions Examples:

More information

Ch. 6 Review. AP Statistics

Ch. 6 Review. AP Statistics Ch. 6 Review AP Statistics 1. The probability of any outcome of a random phenomenon is A) the precise degree of randomness present in the phenomenon. B) any number as long as it is between 0 and 1. C)

More information

Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all

More information

Interlude: Practice Midterm 1

Interlude: Practice Midterm 1 CONDITIONAL PROBABILITY AND INDEPENDENCE 38 Interlude: Practice Midterm 1 This practice exam covers the material from the first four chapters Give yourself 50 minutes to solve the four problems, which

More information

What is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts

What is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts Basic Concepts Introduction to Probability A probability experiment is any experiment whose outcomes relies purely on chance (e.g. throwing a die). It has several possible outcomes, collectively called

More information

Class : XII Mathematics - PROBABILITY. 1. If A and B are two independent events, find P(B) when P(A U B ) = 0.60 and P(A) = 0.35.

Class : XII Mathematics - PROBABILITY. 1. If A and B are two independent events, find P(B) when P(A U B ) = 0.60 and P(A) = 0.35. Class : XII Mathematics - PROBABILITY 1. If A and B are two independent events, find P(B) when P(A U B ) = 0.60 and P(A) = 0.35. 2. A card is drawn from a well shuffled pack of 52 cards. The outcome is

More information

PROBABILITY. Chapter Overview Conditional Probability

PROBABILITY. Chapter Overview Conditional Probability PROBABILITY Chapter. Overview.. Conditional Probability If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the

More information

7.3 Introduction to Probability

7.3 Introduction to Probability 7.3 Introduction to Probability A great many problems that come up in applications of mathematics involve random phenomena - those for which exact prediction is impossible. The best we can do is determine

More information

3 The multiplication rule/miscellaneous counting problems

3 The multiplication rule/miscellaneous counting problems Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,

More information

Math 1320 Chapter Seven Pack. Section 7.1 Sample Spaces and Events. Experiments, Outcomes, and Sample Spaces. Events. Complement of an Event

Math 1320 Chapter Seven Pack. Section 7.1 Sample Spaces and Events. Experiments, Outcomes, and Sample Spaces. Events. Complement of an Event Math 1320 Chapter Seven Pack Section 7.1 Sample Spaces and Events Experiments, Outcomes, and Sample Spaces An experiment is an occurrence with a result, or outcome, that is uncertain before the experiment

More information

Exam. Name. Find the number of subsets of the set. 1) {x x is an even number between 11 and 31} 2) {-13, 0, 13, 14, 15}

Exam. Name. Find the number of subsets of the set. 1) {x x is an even number between 11 and 31} 2) {-13, 0, 13, 14, 15} Exam Name Find the number of subsets of the set. 1) {x x is an even number between 11 and 31} 2) {-13, 0, 13, 1, 15} Let A = 6,, 1, 3, 0, 8, 9. Determine whether the statement is true or false. 3) 9 A

More information

AP Stats - Probability Review

AP Stats - Probability Review AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose

More information

Based on Example 3-2 Illowsky, B., & Dean, S. Collaborative Statistics. Connexions,

Based on Example 3-2 Illowsky, B., & Dean, S. Collaborative Statistics. Connexions, Last Name First Name Class Time Chapter 3-1 Probability: Events and Probabilities PROBABILITY: likelihood or chance that an outcome will happen; long-run relative frequency A probability is a number between

More information

Remember to leave your answers as unreduced fractions.

Remember to leave your answers as unreduced fractions. Probability Worksheet 2 NAME: Remember to leave your answers as unreduced fractions. We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts,

More information

Math227 Homework (Probability Practices) Name

Math227 Homework (Probability Practices) Name Math227 Homework (Probability Practices) Name 1) Use the spinner below to answer the question. Assume that it is equally probable that the pointer will land on any one of the five numbered spaces. If the

More information

number of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.

number of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed. 12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.

More information

Probability. 4.2 More Definitions. 4.3 Some Rules of Probability. Probability Events

Probability. 4.2 More Definitions. 4.3 Some Rules of Probability. Probability Events 1 4 Probability. 4.1 Events Event is an outcome of an experiment or survey. Such as, -rolling a die and turning up six dots, -an individual who votes for the incumbent candidate in an election, -someone

More information

Math 166:505 Fall 2013 Exam 2 - Version A

Math 166:505 Fall 2013 Exam 2 - Version A Name Math 166:505 Fall 2013 Exam 2 - Version A On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work. Signature: Instructions: Part I and II are multiple choice

More information

Section Tree Diagrams. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section Tree Diagrams. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 12.5 Tree Diagrams What You Will Learn Counting Principle Tree Diagrams 12.5-2 Counting Principle If a first experiment can be performed in M distinct ways and a second experiment can be performed

More information

Section 2.2 Sample Space and Events

Section 2.2 Sample Space and Events Section 2.2 Sample Space and Events We consider an experiment whose outcome is not predictable with certainty. However, we suppose that the set of all possible outcomes is known. DEFINITION: The set of

More information

Section 4.2 Exercises

Section 4.2 Exercises Section 4.2 Exercises 1. A company is to elect a secretary and a treasurer from the group of 30 members. How many ways can those two officers be elected? 2. Sandra has 9 shirts and 5 pairs of pants. How

More information

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

More information

8.1 Distributions of Random Variables

8.1 Distributions of Random Variables 8.1 Distributions of Random Variables A random variable is a rule that assigns a number to each outcome of an experiment. We usually denote a random variable by X. There are 3 types of random variables:

More information

Fundamentals of Probability

Fundamentals of Probability Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible

More information

I. WHAT IS PROBABILITY?

I. WHAT IS PROBABILITY? C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

More information

Sample Space: Set of all possible simple outcomes The sample space is Ω (Capital Greek letter Omega). P(Ω) = 1.

Sample Space: Set of all possible simple outcomes The sample space is Ω (Capital Greek letter Omega). P(Ω) = 1. Last Name First Name _Class Time Chapter 3-1 Probability: Events and Probabilities Probability: likelihood or chance that an outcome will happen; long-run relative frequency A probability is a number between

More information

Sample Space, Events, and PROBABILITY

Sample Space, Events, and PROBABILITY Sample Space, Events, and PROBABILITY In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.

More information

Lesson 1: Experimental and Theoretical Probability

Lesson 1: Experimental and Theoretical Probability Lesson 1: Experimental and Theoretical Probability Probability is the study of randomness. For instance, weather is random. In probability, the goal is to determine the chances of certain events happening.

More information

Statistics for Managers Using Microsoft Excel 5th Edition

Statistics for Managers Using Microsoft Excel 5th Edition Statistics for Managers Using Microsoft Excel 5th Edition Chapter 4 Basic Probability Statistics for Managers Using Microsoft Excel, 5e 2008 Pearson Prentice-Hall, Inc. Chap 4-1 Learning Objectives In

More information

Chapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.

Chapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52. Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive

More information

36 Odds, Expected Value, and Conditional Probability

36 Odds, Expected Value, and Conditional Probability 36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face

More information

5.3. Independence and the Multiplication Rule

5.3. Independence and the Multiplication Rule 5.3 Independence and the Multiplication Rule Multiplication Rule The Addition Rule shows how to compute or probabilities P(E or F) under certain conditions The Multiplication Rule shows how to compute

More information

Probability Questions with Solutions

Probability Questions with Solutions Probability Questions with Solutions Tutorial on finding the probability of an event. In what follows, S is the sample space of the experiment in question and E is the event of interest. n(s) is the number

More information

Chapter 15. Probability, continued. Sections 5. Probability spaces with equally likely outcomes

Chapter 15. Probability, continued. Sections 5. Probability spaces with equally likely outcomes Chapter 15. Probability, continued Sections 5. Probability spaces with equally likely outcomes Suppose we have a random experiment with sample space S = {o 1,...,o N }. Suppose the outcomes are equally

More information

Basic Probability. Decision Trees Independence Multiplication Rules Marginal Probability Using the General Multiplication Rule

Basic Probability. Decision Trees Independence Multiplication Rules Marginal Probability Using the General Multiplication Rule 4 Basic Probability USING STATISTICS @ M&R Electronics World 4.1 Basic Probability Concepts Events and Sample Spaces Contingency Tables and Venn Diagrams Simple Probability Joint Probability Marginal Probability

More information

1 Combinations, Permutations, and Elementary Probability

1 Combinations, Permutations, and Elementary Probability 1 Combinations, Permutations, and Elementary Probability Roughly speaking, Permutations are ways of grouping things where the order is important. Combinations are ways of grouping things where the order

More information

Elementary Statistics

Elementary Statistics Elementary Statistics Chapter 03 Dr. hamsary Page 1 Elementary Statistics M. hamsary, Ph.D. Chapter 03 1 Elementary Statistics Chapter 03 Dr. hamsary Page 2 Counting Techniques Multiplication Rule(Principal

More information

Name: Class: Date: 3. What is the probability of a sum of 5 resulting from the roll of two six-sided dice?

Name: Class: Date: 3. What is the probability of a sum of 5 resulting from the roll of two six-sided dice? Name: Class: Date: Sample Distributions Multiple Choice Identify the choice that best completes the statement or answers the question.. Which of the following is not an example of a discrete random variable?

More information

Ch. 3 Probability Unit Assignment

Ch. 3 Probability Unit Assignment Name: Class: _ Date: _ ID: A Ch 3 Probability Unit Assignment Multiple Choice Identify the choice that best completes the statement or answers the question Given the following probabilities, which event

More information

DETERMINING OUTCOMES

DETERMINING OUTCOMES EXAMPLES: DETERMINING OUTCOMES 1. For the experiment of tossing a coin, what are the possible outcomes? 2. For the experiment of tossing two coins, what are the possible outcomes? 3. For the experiment

More information

Basic concepts in probability. Sue Gordon

Basic concepts in probability. Sue Gordon Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

More information

Grade 7/8 Math Circles Fall 2012 Probability

Grade 7/8 Math Circles Fall 2012 Probability 1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics

More information

Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314 Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space

More information

(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.

(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball. Examples for Chapter 3 Probability Math 1040-1 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw

More information

Expected Value Homework Solutions. 1. You roll two dice. What is the probability of two sixes? Of exactly one 6? Of no sixes?

Expected Value Homework Solutions. 1. You roll two dice. What is the probability of two sixes? Of exactly one 6? Of no sixes? Expected Value Homework Solutions 1. You roll two dice. What is the probability of two sixes? Of exactly one 6? Of no sixes? Answer: 1/36, 10/36 = 5/18, 25/36 2. You roll two dice. What is the expected

More information

7.1 Sample space, events, probability

7.1 Sample space, events, probability 7.1 Sample space, events, probability In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.

More information

Math 210 Lecture Notes: Sections Probability

Math 210 Lecture Notes: Sections Probability Math 210 Lecture Notes: Sections 7.1-7.2 Probability Richard Blecksmith Dept. of Mathematical Sciences Northern Illinois University 1. Outcomes of an Experiment A sample space for an experiment is a set

More information

Understanding. Probability and Long-Term Expectations. Chapter 16. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc.

Understanding. Probability and Long-Term Expectations. Chapter 16. Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. Understanding Chapter 16 Probability and Long-Term Expectations Copyright 2005 Brooks/Cole, a division of Thomson Learning, Inc. Thought Question 1: Two very different queries about probability: a. If

More information

Section 6.2 Definition of Probability

Section 6.2 Definition of Probability Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will

More information

May or May Not? Probability & Statistics ANSWER KEYS

May or May Not? Probability & Statistics ANSWER KEYS May or May Not? Probability & Statistics ANSWER KEYS Lessons & Activities for probability 203 Lindsay Perro Clipart by Scrappin Doodles Probability & Statistics Lesson Outline: o Week One Probability Notes

More information

number of equally likely " desired " outcomes numberof " successes " OR

number of equally likely  desired  outcomes numberof  successes  OR Math 107 Probability and Experiments Events or Outcomes in a Sample Space: Probability: Notation: P(event occurring) = numberof waystheevent canoccur total number of equally likely outcomes number of equally

More information

Math 1313 Finite Math Test 4 Supplemental Review

Math 1313 Finite Math Test 4 Supplemental Review Math 1313 Finite Math Test 4 Supplemental Review 1. Let A and B be events in a sample space S. Suppose that P(A) =.45, P(B) =.38, and P( A B) =. 21. Find each of the following: a. P( A B) b. P( B A) c

More information

The Probability of an Event

The Probability of an Event PPENDIX D Counting Principles and Probability D11 D. Probability The Probability of an Event Using Counting Methods to Find Probabilities The Probability of an Event The probability of an event is a number

More information

MEP Y9 Practice Book A

MEP Y9 Practice Book A 6 Probability MEP Y9 Practice Book A 6. The Probability Scale Probabilities are given on a scale of 0 to, as decimals or as fractions; sometimes probabilities are expressed as percentages using a scale

More information

Name: Date: 8.4: Percent Probability. Review: Percentages. Definition a percentage is part of a whole with a denominator of 100.

Name: Date: 8.4: Percent Probability. Review: Percentages. Definition a percentage is part of a whole with a denominator of 100. 1 Name: 2 Name: Date: 8.4: Percent Probability Review: Percentages Definition a percentage is part of a whole with a denominator of 100. Look at 34 100 As a fraction, it is 34 pieces of a whole that was

More information

Chapter Four. Probability. Counting Methods Probability of a Single Event Probability of Multiple Events Expected Value

Chapter Four. Probability. Counting Methods Probability of a Single Event Probability of Multiple Events Expected Value Chapter Four Probability Counting Methods Probability of a Single Event Probability of Multiple Events Expected Value Combinations A combination is a group of selected items. n The number of possible combinations

More information

Exam 1 Review Math 118 All Sections

Exam 1 Review Math 118 All Sections Exam Review Math 8 All Sections This exam will cover sections.-.6 and 2.-2.3 of the textbook. No books, notes, calculators or other aids are allowed on this exam. There is no time limit. It will consist

More information

Probability. Probability. Definition of Probability Theoretical Definition - Analytic View. The probability of event A is:

Probability. Probability. Definition of Probability Theoretical Definition - Analytic View. The probability of event A is: Probability Cal State Northridge Ψ320 Andrew Ainsworth PhD Probability Part of everyday life probability in poker probability in lottery probability of rain Guides our expectations Lies at the heart of

More information

Chapter 7: Sets and Probability Part 1: Sets

Chapter 7: Sets and Probability Part 1: Sets Finite Math : Chapter 7 Notes Sets and Probability 1 7.1 Sets What is a set? Chapter 7: Sets and Probability Part 1: Sets set is a collection of objects. We should always be able to answer the question:

More information

Week in Review #3 (L.1-L.2, )

Week in Review #3 (L.1-L.2, ) Math 166 Week-in-Review - S. Nite 9/15/2012 Page 1 of 11 Week in Review #3 (L.1-L.2, 1.1-1.7) Words/oncepts: conjunction; disjunction; negation; exclusive or ; tautology; contradiction set; elements; roster

More information

7 Probability. Copyright Cengage Learning. All rights reserved.

7 Probability. Copyright Cengage Learning. All rights reserved. 7 Probability Copyright Cengage Learning. All rights reserved. 7.3 Probability and Probability Models Copyright Cengage Learning. All rights reserved. Probability and Probability Models Mathematicians

More information

3. If you were to toss three coins, what are the odds against them all landing heads up? a. 1:8 b. 8:1 c. 7:8 d. 7:1

3. If you were to toss three coins, what are the odds against them all landing heads up? a. 1:8 b. 8:1 c. 7:8 d. 7:1 1/ 5 Name: Answers Score: 0 / 8 (0%) [10 subjective questions not graded] Worksheet: Odds Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1.

More information

Chapter 5: Probability

Chapter 5: Probability Chapter 5: Probability Randomness A random phenomenon is a situation in which we know what outcomes could happen, but we don't know which particular outcome did or will happen However, we can calculate

More information

Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection. 1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

More information

1. How many possible outcomes would there be if three coins were tossed once? A) 2 B) 4 C) 6 D) 8

1. How many possible outcomes would there be if three coins were tossed once? A) 2 B) 4 C) 6 D) 8 Name: Date: 1. How many possible outcomes would there be if three coins were tossed once? A) 2 B) 4 C) 6 D) 8 2. If a sportscaster makes an educated guess as to how well a team will do this season, he

More information

Chapter 5 A Survey of Probability Concepts

Chapter 5 A Survey of Probability Concepts Chapter 5 A Survey of Probability Concepts True/False 1. Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided by the total number of possible

More information

Definition and Calculus of Probability

Definition and Calculus of Probability In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the

More information

0-5 Adding Probabilities. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins.

0-5 Adding Probabilities. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins. a. Copy the table and add a column to show the experimental probability of the spinner landing on each

More information

4.5 Finding Probability Using Tree Diagrams and Outcome Tables

4.5 Finding Probability Using Tree Diagrams and Outcome Tables 4.5 Finding Probability Using ree Diagrams and Outcome ables Games of chance often involve combinations of random events. hese might involve drawing one or more cards from a deck, rolling two dice, or

More information

Probability And Odds Examples

Probability And Odds Examples Probability And Odds Examples. Will the Cubs or the Giants be more likely to win the game? What is the chance of drawing an ace from a deck of cards? What are the possibilities of rain today? What are

More information

Mathematics 426 Robert Gross Homework 3 Answers P(AB) = P(A) + P(B) P(A B) = 0.44 P(AC) = P(A) + P(C) P(A C) = 0.45 P(BC) = P(B) + P(C) P(B C) = 0.

Mathematics 426 Robert Gross Homework 3 Answers P(AB) = P(A) + P(B) P(A B) = 0.44 P(AC) = P(A) + P(C) P(A C) = 0.45 P(BC) = P(B) + P(C) P(B C) = 0. Mathematics 6 Robert Gross Homework 3 Answers 1. Suppose that A, B, and C are events, and you know that PA = 0.7 PA B = 0.78 PA B C = 0.95 PB = 0.50 PB C = 0.8 PC = 0.6 PA C = 0.91 What is PABC? Answer

More information

Review of Probability

Review of Probability Review of Probability Table of Contents Part I: Basic Equations and Notions Sample space Event Mutually exclusive Probability Conditional probability Independence Addition rule Multiplicative rule Using

More information

Events, Sample Space, Probability, and their Properties

Events, Sample Space, Probability, and their Properties 3.1 3.4 Events, Sample Space, Probability, and their Properties Key concepts: events, sample space, probability Probability is a branch of mathematics that deals with modeling of random phenomena or experiments,

More information