Chapter 5 Objectives of chapter: Understand probability values. Know how to determine probability values. Use rules of counting. Section 5-1 Probability Rules What is probability? It s the of the occurrence of some event In plain English, it s the of something happening It also is the foundation for Basic Notation: P = A = P(A) = Probability values must be between and. Scale of probabilities: Different ways to report probabilities: Decimal form Percent form Reduced fraction Page 1 of 46
Experiment some action or process whose outcome cannot be with certainty. Very simple examples: flipping a coin, or rolling a single die Event some specified or a collection of outcomes that may or may not occur when an experiment is performed Sample space list of all possible for the experiment. Probability model list of all the possible outcomes of a probability experiment, and each outcome s. Note that the sum of the probabilities of all outcomes must equal. Unusual event an event that has a probability of occurring. Typically (but not always), an event with a probability is considered to be unusual. A single coin toss Probability Model: Sample space = Outcome Probability Toss two coins: Sample space = Rolling a single die Sample space = Rolling two dice Page 2 of 46
Three methods to define the probability of an event: 1. The Empirical Method (aka Experimental) 2. The Classical Method (aka Theoretical) 3. The Subjective Method 1. Empirical Method (Experimental Probability) Empirical evidence is evidence based on the outcomes of an From the M146 class survey: Dominant Hand Frequency Dominant Hand Probability Right Left Right Left Roll two dice 100 times. Record the number of times you get exactly one 6: Estimate the probability of the event using the Empirical Approach: Page 3 of 46
2. Classical Method (Theoretical Probability) The classical method calculates the probability that is by mathematics. It requires all of the outcomes for the experiment to be to occur. Examples: All of the experiments listed on page 2 If you roll a single die, what is the probability of getting an even number? P(even) = number of ways it can occur total number of outcomes If you roll two dice, what is the probability of getting exactly one six? P(one 6) = number of ways it can occur total number of outcomes Page 4 of 46
If you roll two dice, what is the probability of getting exactly one six? Experimental probability = Theoretical probability = Experiment Roll two dice 1000 times. Record the number of times you get exactly one 6. Law of Large Numbers: States that if an experiment has a of trials, The experimental probability will the theoretical probability or the probability predicted by mathematics. Therefore, the estimate gets with more trials. 3. Subjective Method Basically an. Can base on past experience and current knowledge of relevant circumstances. What is the probability that your car will not start when you try to leave campus? What is the probability that the Mariners will win the World Series this season? Page 5 of 46
Tree Diagrams Useful for small problems to list the sample space Flip a coin (record H or T) Roll a single die (record 1, 2, 3, 4, 5, or 6) Define event as: A = T, even number Calculate experimental probability: Number of T/even = Total no. of trials = Experimental probability of A = Calculate theoretical probability of T/even by listing the sample space: Start: Page 6 of 46
1. Identifying Probability Values a. What is the probability of an event that is certain to occur? b. What is the probability of an impossible event? c. A sample space consists of 10 separate events that are equally likely. What is the probability of each? d. On a true/false test, what is the probability of answering a question correctly if you make a random guess? e. On a multiple-choice test with five possible answers for each questions, what is the probability of answering a question correctly if you make a random guess? 2. Excluding leap years, and assuming each birthday is equally likely, what is the probability that a randomly selected person has a birthday on the 1st day of a month? 3. What is the probability of rolling a pair of dice and obtaining a total score of 10 or more? (Hint: look at the sample space on p. 2). 4. Page 7 of 46
Section 5.2 The Addition Rule and Complements Objectives: The Addition Rule for Disjoint Events The General Addition Rule Complement Rule Disjoint (or Mutually Exclusive) Events Disjoint events have. There is at all. Rolling two dice A = event that the black die is a 1 B = event that both dice are displaying the same number C = event that the sum of the dice is more than 7 1. A & B: are they disjoint? 2. A & C: are they disjoint? Page 8 of 46
Addition Rule for Disjoint Events The special addition rule only applies to events. Calculate the probability of either event A happening as follows: event B happening P(A or B) = P(A or B or C) = Christmas ornaments (never too early to start shopping!) Define the following events: A = plain round ornament B = pointy decorated ornament If one ornament is randomly selected, find the probability that it is a plain round ornament or a pointy decorated ornament: P(A or B) = P(plain round or pointy decorated) = Page 9 of 46
Religion in America Find the probability that a randomly selected American adult is Catholic or Protestant. P(Catholic or Protestant) = Page 10 of 46
General Addition Rule The general addition rule is for events that are mutually exclusive. P(A or B) = Christmas ornaments Define the following events: A = round ornament B = decorated ornament If one ornament is randomly selected, find the probability that it is a round ornament or a decorated ornament: First, solve intuitively, by just looking at the picture: P(A or B) = P(round or decorated) = Second, solve rigorously, by applying the General Addition Rule: P(A or B) = Key Point: associate the word or with probabilities of the Page 11 of 46
Using the General Addition Rule with Contingency Tables A contingency table, or two-way table is used to record and analyze the relationship between two or more variables, usually categorical variables The results of the sinking of the Titanic, which had a total of 2223 passengers Men Women Boys Girls Total Survived 332 318 29 27 706 Did not survive 1360 104 35 18 1517 Total 1692 422 64 45 2223 Row variable is: Column variable is: 1. Determine the probability that a randomly selected passenger is a woman. 2. Determine the probability that a randomly selected passenger is a boy or a girl. 3. Determine the probability that a randomly selected passenger is a man or someone who survived the sinking. Page 12 of 46
Complement of an Event Every event has a complement event, which is basically the of the event. Event A: Complement of A: Notation for complement: Roll a single die A = roll a 6 A c = Use a Venn Diagram to show the relationship between event A and its complement: Complement Rule Each of the events has an associated probability: P(A) = probability that P(A c ) = probability that A Relationship between these two probabilities is: Rewrite this into the Complement Rule: Page 13 of 46
The complement rule can be useful for simplifying calculations. Find the probability that the religious affiliation of a randomly selected US adult is not Jewish. P(not Jewish) = Page 14 of 46
Here is a standard deck of playing cards, 52 cards total, with 4 suits (spades, hearts, clubs and diamonds). Hearts and diamonds are red, spades and clubs are black. Each suit goes from ace, 2, 3,, up through Jack, Queen and King, where Jack, Queen and King are considered to be face cards. 1. A card is drawn at random from a deck. What is the probability that it is an ace or a king? 2. A card is drawn at random from a deck. What is the probability that it is either a red card or an ace? 3. A card is drawn at random from a deck. What is the probability that it is NOT an ace or a king? Page 15 of 46
4. A couple is planning on having three children. What is the probability that they have three of the same gender? (Hint: try a tree diagram). 5. Page 16 of 46
Section 5.3 Independence and the Multiplication Rule 1. Identify Independent Events 2. Multiplication Rule for Independent Events 3. Computing at least probabilities Independence Two events are independent if the knowledge that one event occurred does of the other event occurring. Two events are dependent if the occurrence of one event the probability of another event. Examples: 1. Independent events 2. Blocks, 4 squares, 3 triangles 1. If I randomly select one, what is the probability of selecting a square? 2. Assuming that I got a square on the first grab, what is the probability that I reach in a second time and grab a triangle? It! If I : P(triangle) = If I : P(triangle) = Page 17 of 46
Key Point: If sampling is done replacement, then the events are. If sampling is done replacement, then the events are. IF the events are, then can use the Multiplication Rule for Independent Events to calculate the probability of two events happening. Multiplication Rule for Independent Events If A and B are independent events, then: P(A and B) = This can be extended to multiple independent events: P(A and B and C and ) = Notice that this applies to. In other words: Event A occurs in, Then Event B occurs in (and possibly more events) Roll one die, then a second die. What is the probability of getting a 1 on both? A = get a 1 on first die B = get a 1 on second die Page 18 of 46
If a fair die (singular of dice) is rolled five times, which of the following ordered sequences of results, if any, is MOST LIKELY to occur? a. 3 5 1 6 2 b. 4 2 6 1 5 c. 2 2 2 2 2 d. Sequences (a) and (b) are equally likely. e. All of the above sequences are equally likely. Key Point: associate the word and with of the probabilities. Christmas lights are often designed with a series circuit. This means that when one light burns out the entire string of lights goes black. Suppose that the lights are designed so that the probability a bulb will last 2 years is 0.995. The success or failure of a bulb is independent of the success or failure of other bulbs. What is the probability that in a string of 100 lights all 100 will last 2 years? Computing At-Least Probabilities Complement Rule: If A = at least one of something happens, then A c = For the Christmas lights, what is the probability that at least one bulb will burn out in 2 years? Page 19 of 46
2016 Presidential Election Exit Poll Results An exit poll was conducted by Edison Research during the 2016 U.S. Presidential election, and is based on questionnaires completed by voters leaving 350 voting places throughout the US, and also including telephone interviews with early and absentee voters. The following table provides the results, by race of the voters. Voted for Clinton Voted for Trump Voted for Other/No Answer White 37% 57% 6% Black 89% 8% 3% Latino 66% 28% 6% Asian 65% 27% 8% Other 56% 36% 8% Sources: http://www.cnn.com/election/results/exit-polls/national/president https://www.nytimes.com/interactive/2016/11/08/us/politics/election-exit-polls.html?_r=0 Calculate the following probabilities to three significant figures. 1. If two White voters are randomly selected, what is the probability that they both voted for Trump? 2. If three Black voters are randomly selected, what is the probability that all three of them voted for Clinton? 3. If two Asian voters are randomly selected, what is the probability that the first voted for Clinton and the second voted for Trump? 4. If three Latino voters are randomly selected, what is the probability that at least one of them voted for Clinton? Page 20 of 46
Section 5.4 Conditional Probability and the General Multiplication Rule 1. Compute Conditional Probabilities 2. Compute probabilities using the General Multiplication Rule The conditional probability of an event is the probability that the event occurs, assuming that another event has. The conditional probability that event B occurs given that event A has occurred is written: Face cards Let: A = get a face card (Jack, Queen or King) B = get a Queen a. If one card is randomly selected, find the probability that it is a Queen. b. If one card is randomly selected, find the probability that it is a Queen, given that it is a face card. Conclusion: Knowing that it is a face card it is a Queen. the probability that Page 21 of 46
Conditional Probability Rule: If A and B are any two events, then: P(B A) = Titanic passengers, 2223 total Men Women Boys Girls Total Survived 332 318 29 27 706 Did not survive 1360 104 35 18 1517 Total 1692 422 64 45 2223 Assume that one of the 2223 passengers is randomly selected. a. Determine the probability that the passenger is a man. b. Determine the probability that the passenger is a man, given that the selected passenger did not survive. Conclusion: Knowing that the passenger did not survive probability that it is a man. the Page 22 of 46
Defining Independent Events using Conditional Probabilities From the Titanic example on the previous page: P(B) = P(man) = P(B A) = P(man did not survive) = Are the events man and did not survive independent? Page 23 of 46
The International Shark Attack File, maintained by the American Elasmobranch Society and the Florida Museum of Natural History, is a compilation of all known shark attacks around the globe from the mid 1500s to the present. Following is a contingency table providing a cross-classification of worldwide reported shark attacks during the 1900s, by country and lethality of attack. a. Find the probability that an attack occurred in the United States. b. Find the probability that an attack occurred in the United States and that it was fatal. c. Find the probability that an attack was fatal. d. Find the probability that an attack was fatal, given that it occurred in the United States. e. Find the probability that an attack occurred in the United States, given that it was fatal. f. Are the events fatal and occurred in the United States independent? Page 24 of 46
IF the events are, then use the General Multiplication Rule to calculate the probability of two events happening. General Multiplication Rule If A and B are any two events, then: P(A and B) = Blocks, 4 squares, 3 triangles 1. Assuming replacement after each selection, find the probability that I randomly select two blocks (one at a time), and get a square first and a triangle second: P(square & triangle) = Note: in this case, the events are independent. 2. Assuming replacement after each selection, find the probability that I randomly select two blocks (one at a time), and get a square first and a triangle second: P(square & triangle) = Note: in this case, the events are NOT independent. From a deck of cards, find the probability of selecting two cards without replacement, and having them both be Kings. Pick two cards from a deck without replacement: What is the probability of getting those two cards? Page 25 of 46
Often, we assume sampling for calculations, even though technically the sampling is done without replacement. The reason: calculating probabilities WITH replacement is much easier, because we don t have to worry about the conditional probabilities changing every time we make a selection. CBC students Assume 7400 students total, 4144 female students Question: If one student is selected at random, what is the probability that it is a female student? Question: If five different students are selected at random, what is the probability that ALL FIVE are female students? Calculate without replacement: P(female & female & female & female & female) = Now, calculate assuming with replacement: It s OK in this case to assume replacement, because the sample size is very compared to the population size. Page 26 of 46
Using the Complement Rule CBC students If five different students are randomly selected, what is the probability of selecting at least one student who is female? Remember, just calculated that P(female) = A = A c = Page 27 of 46
1. Two cards are drawn from a deck without replacement. What is the probability they are both diamonds? 2. 3. Assume that you are going to (maybe this weekend) take a quiz with 5 multiple choice questions, each with 4 possible answers. You randomly guess. What is the probability of getting at least one question right? (note: please do not actually use this strategy!) Page 28 of 46
4. Use the sample data from the following table, which includes results from experiments conducted with 100 subjects, each of whom was given a polygraph test. Polygraph Indicated Truth Polygraph Indicated Lie Subject actually told the truth 65 15 Subject actually told a lie 3 17 a. If 1 of the 100 subjects is randomly selected, find the probability of getting someone who told the truth or had the polygraph test indicate that the truth was being told. b. If two different subjects are randomly selected, find the probability that they both had the polygraph test indicate that a lie was being told. Do the calculation without replacement. c. Repeat the calculation in part b., but this time assume replacement. Page 29 of 46
The Birthday Problem Basic idea is to find the probability for our class that AT LEAST TWO people have the same birthday. Complement: people in the room have the same birthday. Assumptions: Assume 365 possible birthdays Assume all birthdays are equally likely Assume no twins (or otherwise) in the room ( ) Source: Tri-City Herald, October 6, 2007 Page 30 of 46
Use the Complement Rule: It s easier to calculate the probability that NO TWO people in the class have the same birthday. A = at least two people have the same birthday (what we want) A c = nobody has the same birthday P(A) = 1 P(A c ) P(at least 2 people have same birthday) = 1 P(nobody has same birthday) Calculate: P(nobody has the same birthday), or P(A c ) 1. Find the probability that TWO randomly selected people do NOT have the same birthday: Probability that the 1 st person has a birthday = Probability that the 2 nd person s birthday is different = P(1 st birthday AND different 2 nd birthday) = 2. Find the probability that out of THREE randomly selected people, NONE of them have the same birthday. Probability that the 3 rd person s birthday is different from the first two = P(1 st birthday AND different 2 nd birthday AND different 3 rd birthday) = Page 31 of 46
Number of people in the room: r = 3. Find the probability that out of randomly selected people, NONE of them have the same birthday. Probability that the person s birthday is different from the first = P(1 st birthday AND different 2 nd birthday AND different 3 rd birthday AND different birthday) = P(A c ) = (probability that nobody in here has same b-day) Now take the complement of this value: Therefore, P(A) = P(at least two people in here have the same birthday) = 1 P(A c ) = Page 32 of 46
Easier way: the probability that out of r randomly selected people, NONE of them have the same birthday, can be represented mathematically by the formula: Probability = where, r = no. of people 365Pr = no. of permutations of 365 objects taken r at a time To calculate: 365Pr Calculator Commands TI-30X 365 2 nd x y _r_ 2 nd npr = TI-30Xa 365 2 nd npr _r_ = TI-30XIIb TI-30XIIs 365 PRB npr (enter) _r_ = TI-34II 365 PRB _r_ = TI-36X 365 x y _r_ 2 nd npr = TI-68 365 3 rd npr _r_ = Casio fx-260 solar 365 SHIFT npr _r_ = Test: 365P2 = 132,860 Try to use your calculator to calculate the value we just computed. 365 P r = r = Calculate r 365 P(at least two people in here have the same birthday) = 1 365 P r = r 365 Page 33 of 46
Section 5.5 Counting Techniques The principal of counting means: finding how many different an event can have. Powerball January 2016 record Powerball drawing: $1.6 billion grand prize If you buy a Powerball ticket, what is the probability that you will have the winning numbers? To calculate this, we have to know how many different ways there are to choose the numbers. How many outcomes when you: Roll a single die Roll two dice Draw a single card Toss two coins For a couple having 3 children Techniques to count: Page 34 of 46
The Multiplication Rule (Basic Counting Rule) Talking about making a sequence of choices from separate categories. In other words, sequential trials, or sequential events: Pick something from the first category, Then pick from the second category, Etc. Each selection from each category can have a certain number of outcomes: Category No. of outcomes The Multiplication Rule says that: To find the number of possible outcomes, together the individual number of outcomes for each category. Flipping two coins Total outcomes = Rolling two dice Total outcomes = A couple having three kids Total outcomes = Page 35 of 46
Assume that a license plate consists of 4 letters followed by three digits. How many plates are possible (letters and digits may be repeated)? Event Pick 1 st letter No. of outcomes Pick 2 nd letter Pick 3 rd letter Pick 4 th letter Pick 1 st digit Pick 2 nd digit Pick 3 rd digit Total no. of outcomes = Applying Counting Rules to Probability What s the probability of randomly generating 4 letters and 3 digits and having it be your plate? The outcomes are equally likely, so apply the Classical Method to calculate probabilities: P = How many plates are possible if the letters and digits may NOT be repeated? Page 36 of 46
Factorial Notation n! is factorial notation, and it is read n factorial. n! = n must be a non-negative integer Examples: Permutations A permutation is: any different of a certain number of objects. KEY POINT: matters when you are counting up permutations! How many different ways can the objects be arranged? Page 37 of 46
How many different ways can two objects at a time selected from this group be arranged? Permutation Rule: npr = npr = number of permutations n = total number of objects r = number of them taken at a time Read as: the number of permutations of n objects taken r at a time (example this page) 4P2 = (example previous page) 3P3 = Special Permutations Rule: Just a special case of the permutation rule. A collection of n different items can be arranged in order ways. Page 38 of 46
Permutations with Nondistinct Items How many ways to arrange all of the objects: (they are ALL distinct) How many ways to arrange all of the objects: (they are NOT all distinct) Will it be more, or less, or the same? Formula: The number of permutations of n objects of which n1 are of one kind, n2 are of a second kind,, and nk are of a k th kind is given by: How many different six-digit numerals can be written using all of the following six digits: 4, 4, 5, 5, 5, 7? Page 39 of 46
Combinations Combinations are different from permutations because of the objects does not matter Not how many different arrangements there are, just how many different. a, b, c Previously found that there were permutations of these objects. How many different combinations of these objects are there? a, b, c, d Previously found that there were objects at a time from this group. permutations when selecting two How many different combinations of two objects can be selected from this group? Combinations Rule: ncr = ncr = number of combinations n = total number of objects r = number of them taken at a time Read as: the number of combinations of n objects taken r at a time Find the number of combinations of 25 objects taken 8 at a time. Page 40 of 46
Selections from Two (or more) Subgroups Cracked Eggs A carton contains 12 eggs, 3 of which are cracked. If we randomly select 5 of the eggs for hard boiling, how many outcomes are there for the following events? a. Select any 5 of the eggs. b. All of the cracked eggs are selected. Note: in this case, we are choosing specific numbers from the two sub-groups, cracked and not cracked. 1. Use the Rule to determine the number of outcomes for the selections from each subgroup. 2. Use the Rule to multiply the individual outcomes together. Cracked: 3 Not cracked: 9 Select: Select: (for a total of 5) What is the probability that all of the cracked eggs are selected? Page 41 of 46
Powerball If you buy a Powerball ticket, how many different outcomes are there? In other words, how many different ways to choose the numbers? Draw 5 different white balls out of a drum that has 69 white balls and 1 red ball out of a drum that has 26 red balls in it. White: 69 Red: 26 Select: Select: What is the probability that you win? Page 42 of 46
Summary of Counting Methods Sequentially, from different categories Items Selected Two main questions: 1. Is the selection with or without replacement? 2. Does order matter? With Replacement Without Replacement From one big bag of items Multiplication Rule How many total items are there? = n How many of them are being selected? = r Order does matter Order does NOT matter Special Permutations Rule (when some items are identical to others) no Multiplication Rule All items distinct? yes Permutations Combinations Special case: if you are selecting or drawing from 2 or more groups of things, calculate the combinations for each and multiply together using the Multiplication Rule. Note that these two are equivalent (when counting without replacement), so you can use either one (I usually do permutations). Page 43 of 46
Example 1: From a committee of 8 people, how many ways can we choose a subcommittee of 2 people? 1. With or without replacement? 2. Does order matter? Example 2: From a committee of 8 people, how many ways can we choose a chairperson and a vice chair? 1. With or without replacement? 2. Does order matter? Example 3: How many 5-card hands can be dealt from a deck of 52 cards? 1. With or without replacement? 2. Does order matter? Page 44 of 46
1. How many ways can a 10-question multiple choice test be answered if there are 5 possible answers (A, B, C, D, and E) to each question? Hint: this is kind of like the license plate problem! 2. Ten people gather for a meeting. If each person shakes hands with each other person exactly once, what is the total number of handshakes? 3. In the Washington State Lotto game, players pick six different numbers between 1 and 49. What is the probability of winning the Lotto? 4. A nurse has 6 patients to visit. How many different ways can he make his rounds if he visits each patient only once? Page 45 of 46
5. In a dog show, a German Shepherd is supposed to pick the correct two objects from a set of 20 objects. In how many ways can the dog pick two objects? 6. A slot machine consists of three wheels with 12 different objects on a wheel (each wheel has the same 12 objects). How many different outcomes are possible? 7. In a Jumble puzzle, you are supposed to unscramble letters to form words. How many ways can the letters CATSITTISS be arranged? (Also, what is the word?) 8. The Hazelwood city council consists of 5 men and 4 women. How many different subcommittees can be formed that consist of 3 men and 2 women? Page 46 of 46