Note: In this chapter, we only cover sections 4-1 through 4-3

Basic Concepts of Probability Sections 4-1 & 4-2 Basic Concepts of Probability

Inferential Statistics Because inferential statistics involves using information obtained from part of a population (a sample) to draw conclusions about the entire population, we can never be certain that our conclusions are correct --- uncertainty is inherent in inferential statistics. So before we can understand, develop, and apply the methods of inferential statistics, we need to become familiar with uncertainty. We can’t be sure something will happen, but we can find the probability that it will happen.

Inferential Statistics Statisticians think in probabilities. If you tell them something happened to you that has an extremely low probability, they won’t believe you. We see this on shows like CSI or Law and Order all the time. The lawyer says to the expert witness, “So you’re telling me that it’s possible my client didn’t do it.” The scientist replies, “Well, I suppose it’s possible, but the probability is so small that I really doubt it.” Statisticians rely on reasonable doubt. The alternative theory could have happened, but they are sure beyond a reasonable doubt that it didn’t. We don’t care if it’s possible, but if it’s probable.

Definitions Event Any collection of one or more outcomes. Simple Event Only one outcome, cannot be broken into smaller parts. Sample Space Consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further in an experiment. Usually denoted by a capital S.

Simple Event vs. Event When rolling a single die, rolling a 5 is a simple event because there is only one way it can happen. When rolling two dice, rolling a 5 is not a simple event because it can happen more than one way, such as 1-4 and 2-3. Sometimes we are told how many outcomes are possible, and sometimes we have to figure out the sample space.

Sample Space Examples S = {1, 2, 3, 4, 5, 6} A fair six-sided die is rolled. List the sample space for this random experiment. S = {1, 2, 3, 4, 5, 6} This is because there is a possibility of rolling a 1, 2, 3, 4, 5, or 6 when tossing the die.

Sample Space Example List the sample space for a two-child family. At times, it is better to draw a picture. Let B represent the outcome of a boy and G for a girl. See the tree diagram below. Outcomes B BB B G BG GB B G First Child G GG Second Child S = {BB, BG, GB, GG}

P (A) denotes the probability of event A occurring. Notation for Probabilities P (A) denotes the probability of event A occurring.

Example If you are wanting to find the probability of having two girls in a family of two, possible ways to denote that would be: P(GG) or P(2 girls) Meaning “Find the probability of two girls”.

Probabilities Probabilities are values between 0 and 1 (often percentages written as decimals) P(A) = 0 is a probability of 0. This is a 0% chance, so it is impossible. P(A) = 1 is a probability of 1. This is a 100% chance, so it is certain to happen. P(A) = 0 No chance P(A) = 0.5 50-50 chance P(A) = 1 Certain

Identifying Unusual Results From Probabilities With probabilities, we consider an event unusual if its probability of occurring is very low. What counts as “very low” is not set in stone, but we often consider it to be 0.05 or less. So an event is unusual if there is less than a 5% chance that it will happen. An event is considered rare if its probability is less than 0.001, or 0.1%.

Rounding Off Probabilities When writing a probability, either use a fraction or a decimal rounded to three significant digits. Rounding to less than this gives inaccurate and misleading probabilities. Note: Zeros at the beginning or end of a number are not considered to be significant. Zeros in the middle of numbers are significant.

Rounding Off Probabilities This is different from rounding to 3 decimal places. To round to 3 significant digits, reading left to right, start counting three places at the first nonzero digit. 2/3 = 0.6666666667 = 0.667 0.0040823 = 0.00408 432/7842 = 0.055087987 = 0.0551

There are three methods for computing probabilities, used in different situations. Relative Frequency Approximation Classical Approach for Equally Likely Outcomes Subjective Probability

Computing Probability Basic Rules for Computing Probability Relative Frequency Approximation of Probability Conduct a statistical experiment or read about actual data, and count the number of times event A actually occurs. Based on these actual results, P(A) is estimated as follows: P(A) = number of times A occurred number of times trial was repeated

Remember P(A) stands for “the probability of A” We use this rule when events are not equally likely to happen, or when all we have to work with is sample data. In these cases, you will be given data to use.

Law of Large Numbers The more times you repeat an experiment, or the more people in a study, the more accurate your estimated probabilities will be.

See if you can do this on your own first. See next slide for solution Example Estimate the probability of a 25-year old man dying before he turns 26, if out of 100,000 men aged 25, 138 died within the next year. See if you can do this on your own first. See next slide for solution

Key Pieces of Information The event that we are interested in is “a 25 year old man dying before he turns 26”, so A = a 25-year old man dying before he turns 26 The number of times the event has occurred is 138 times The number of the times the trial is repeated is 100,000

Solution The probability of a 25-year old man dying before he turned 26 is 138 chances out of 100,000 or approximately 0.00138 or approximately 0.138% (which of course means that the probability of a 25-year old man dying before he turned 26 is very slim) Note: This is an estimate for what the true probability is. With larger values of n, this estimate will become closer to the true value.

See if you can do this on your own first! See next slide for solution Example Of the first 42 presidents of the United States, 26 were lawyers. What is the probability of randomly selecting from these 42 presidents a president who was a lawyer? See if you can do this on your own first! See next slide for solution

Key Pieces of Information The event that we are interested in is “a president being a lawyer”, so A = a president being a lawyer. The number of times the event has occurred is 26 times The number of the times the trial is repeated (number of presidents studied) is 42

Solution The probability of a president being a lawyer is approximately 0.619 (rounded to 3 significant digits) or approximately 61.9%

Example If a quality control test shows that there are 5 defective printers and there are 15 that are good, find the probability of randomly selecting one printer that is defective. Think about the problem a little bit, what would you do? See next slide for solution

Key Pieces of Information If a quality control test shows that there are 5 defective printers and there are 15 that are good, find the probability of randomly selecting one printer that is defective. We know we have 5 defective printers We have a total of 20 printers (5 defective and 15 good ones) Therefore, we have a 5 in 20 chance of selecting a printer that is defective.

There is a 25% chance we will select a defective printer. Solution A = selecting one printer that is defective There is a 25% chance we will select a defective printer.

Computing Probability Basic Rules for Computing Probability Classical Approach to Probability for Equally Likely Outcomes Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. If event A can occur in f of these n ways, then P(A) = number of ways A can occur number of different simple events f n =

See next slide for solution Example If a two-child family is selected at random, what is the probability of two boys? See if you can do this on your own first! Look back at our tree diagram for help. See next slide for solution

Key Pieces of Information We can list the sample space from our earlier slide S = {BB, GB, BG, GG} Note: We always assume that different order means different event. The event that we are interested in is “having two boys”, so A = two boys. The number of ways A can occur is only one. Two boys occurs once in our sample space. The number of simple events is four. Our sample space consists of four different options.

Solution The probability of having two boys in a two-child family is approximately 25%.

Example Use the following sample space of equally likely events to answer the questions: S = {AABA, ABAC, CAAB, CCBB, BACA, BBBB, BCBC, AAAA} a.) What is the probability of getting an event with at least 2 “A”’s? b.) What is the probability of getting an event with no “C’s”? See if you can do these on your own first! See next slides for solutions

Part A Key Pieces of Information The event that we are interested in is “getting an event with at least 2 “A”s”, so A = getting an event with at least 2 “A”s. Remember at least two means “2 or more”. If you look at the sample space, you see this can occur 5 ways. The number of simple events is 8.

You could state it in decimal or percent form as well. Part A Solution The probability of getting an event with at least 2 “A”s is 5 out of 8. You could state it in decimal or percent form as well.

Part B Key Pieces of Information The event (something that may or may not occur) that we are interested in is “getting an event with no C’s”, so A = getting an event with no C’s. If you look at the sample space, you see this can occur in 3 ways. The number of simple events is 8.

Part B Solution The probability of getting an event with no C’s is 3 out of 8. You could state it in decimal or percent form as well.

Computing Probability Basic Rules for Computing Probability Subjective Probabilities The probability of an event is estimated based on knowledge of the relevant circumstances and a little bit of guesswork.

Subjective probability is a measure of belief and can be based on past experience. A good example would be the predictions the weatherman makes about the weather. He/She might say it will rain tomorrow, but we have no statistical data telling us that it will rain for sure. We can look at past weather that has happened before on that day, but we cannot be exactly sure. We will not be working with subjective probabilities in this class.

Section 4-3 Addition Rule

P(A or B) = P (A occurs or B occurs or both occur) Definition Compound Event Any event combining 2 or more simple events P(A or B) = P (A occurs or B occurs or both occur)

Definition Events A and B are disjoint (or mutually exclusive) if they cannot both occur together. A and B A B A B Not Disjoint Disjoint

Mutually Exclusive You pull a card from a standard deck. A = pulling a spade B = pulling a heart These events are mutually exclusive, because if you pull a spade, it cannot also be a heart.

Not Mutually Exclusive You pull a card from a standard deck. A = pulling a spade B = pulling a 3 These events are not mutually exclusive, because it is possible to pull a 3 of spades.

General Rule for a Compound Event In this section, we are interested in P(A or B), which is the probability that A, B, or both occur. When finding the probability that event A occurs or event B occurs, find the total number of ways A can occur and the total number of ways B can occur, but be sure that no outcome is counted more than once.

Compound Event Addition Rule If the events are not mutually exclusive, then adding the probabilities of A and B will count the overlap twice. This formula subtracts the extra.

Example People and Their Cars Men Women Total Car 10 8 18 Truck 8 4 12 Van/SUV 3 6 9 Total 21 18 39 I stop people on the way out of the grocery store and ask them what they drive.

Example People and Their Cars Men Women Total Car 10 8 18 Truck 8 4 12 Van/SUV 3 6 9 Total 21 18 39 If you randomly select someone from my list, find the probability that they drive a truck or van/SUV.

Solution People and Their Cars Men Women Total Car 10 8 18 Truck 8 4 12 Van/SUV 3 6 9 Total 21 18 39 Are the events mutually exclusive? They are, so there is no overlap to worry about.

Example People and Their Cars Men Women Total Car 10 8 18 Truck 8 4 12 Van/SUV 3 6 9 Total 21 18 39 If you randomly select someone from my list, find the probability that they are a man or drive a car.

Solution People and Their Cars Men Women Total Car 10 8 18 Truck 8 4 12 Van/SUV 3 6 9 Total 21 18 39 Are the events mutually exclusive? Here they are not, which means if you add the totals for each you count some twice (the 10 people who are men and drive cars).

Alternate Solution People and Their Cars Men Women Total Car 10 8 18 Truck 8 4 12 Van/SUV 3 6 9 Total 21 18 39 Another way to solve this is to add up the relevant individual values (since we have them), rather than the totals. The individual values we have circled are 8 + 10 + 8 + 3 = 29.

Definition The complement of event A, denoted by , is when the event A does not occur. For instance: If the trial of an experiment is rolling a die, then “rolling a 1” (event A) and “not rolling a 1” (event ) are complements. If the trial of an experiment is having a child, then “having a girl” and “having a boy” are complements. (Having a boy is the same as not having a girl.)

Rule of Complementary Events This rule basically says that the probability of an event occurring is 1 minus the probability that it doesn’t occur. If there is a 30% chance of rain today (so P(A) = 0.3), then there is a 70% chance it will not rain today (so P( ) = 0.7).

Example In reality, when a baby is born, P(boy) = 0.512. Find Since we know the probability of having a boy is 0.512, we can easily find the probability of not having a boy

See next slide for final solution. Example If someone is randomly selected, find the probability that his or her birthday is not in December. Keep in mind that each month is a different length, so we need to use the number of days. We have a total of 365 days in a year. We know the probability of having a birthday is December is 31 out of 365. Hence, if we subtract that fraction from one, we can find the probability of not having a birthday in December. See next slide for final solution.

(Notice it is rounded to three significant digits!) Solution (Notice it is rounded to three significant digits!) The probability of having a birthday not in December is approximately 91.5%.