Intro to Statistics, Statistics 1034. Chapter 4 Notes
Intro to Statistics, Statistics 1034. Chapter 4 Notes Stat 1034
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This 9 page Class Notes was uploaded by Alyssa Notetaker on Wednesday February 10, 2016. The Class Notes belongs to Stat 1034 at University of Cincinnati taught by Sarah Myers in Spring 2016. Since its upload, it has received 15 views. For similar materials see Elementary Statistics I in Statistics at University of Cincinnati.
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Date Created: 02/10/16
Chapter 4 Elementary Probability Theory Material Extracted From Textbook (Brase, Charles Henry., and Corrinne Pellillo. Brase. Understandable Statistics: Concepts and Methods . 11th ed. N.p.: Cengage Learning, n.d. Print.) 4.1 What is Probability? Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will occur. Probabilities closer to 1 indicate that the event is more likely to occur. Probabilities closer to 0 indicate that the event is less likely to occur. Probability Assignments 1. A probability assignment based on intuition incorporates past experience, judgement, or opinion to estimate the likelihood of an event . 2. A probability assignment based on relative frequency uses the formula: Probability of event = relative frequency = f/n Where f is the frequency of the event occurrence in a sample of n observations. 3. A probability assignment based on equally likely outcomes uses the formula: Probability of event = # of outcomes favorable to event / total # of outcomes Law of Large Numbers: In the long run, as the sample size increases and increases, the relative frequencies of outcomes get closer and closer to the theoretical (or actual) probability value. ● The underlying assumption we make is that if events occurred a certain percentage of times in the past, they will occur about the same percentage of times in the future. Statistical Experiment/Statistical Observation: Can be thought of as any random activity that results in a definite outcome. Event: Is a collection of one or more outcomes of a statistical experiment or observation. Simple Event: Is one particular outcome of a statistical experiment. Sample Space: The set of all simple events. ● The sum of the probabilities of all simple events in a sample space must equal 1. Interpreting Probabilities: ● The closer the probability is to 1, the more likely the event is to occur. ○ Just because the event of a probability is high, it is not certainty that the event will occur. ● Similarly, if the likelihood of an event is low, it is possible that the event might occur. Events with low probability but big consequences are of special concern. ● Some of people’s biggest mistakes in a person’s life can result from either misjudging: a) the size of an event’s impact. b) the likelihood the event will occur. ● An event of great importance cannot be ignored even if it has a low probability of occurrence. What Does the Probability of an Event Tell Us? ● The probability of an event A tells us the likelihood that event A will occur. If the probability is 1, the event A is certain to occur. If the probability is 0, the event A will not occur. ● The probability of event A applies only in the context of conditions surrounding the sample space containing event A. ● If we know the probability of event A, then we can easily compute the probability of event not A in the context of the same sample space. P(notA)= 1P(A). Probability Related to Statistics ● If probability did not exist, then inferential statistics would not exist. ● Probability you know the overall description of the population. The central problem is to compute the likelihood of a specific outcome. ● Statistics you know only the result of a sample drawn from a statistic. 4.2 Some Probability Rules Compound Events Conditional Probability and Multiplication Rules: Independent Events: Two events are independent if the occurrence or nonoccurrence of one event does not change the probability that the other event will occur. Dependent Events: Two events are dependent if the occurrence or nonoccurrence of one event changes the probability that the other event will occur. Why Does the Independence or Dependence Matter? ● The type of the events determines the way we compute the probability of the two events happening together. Multiplication for Independent Events P(A and B) = P(A) x P(B) Multiplication for Dependent Events P(A and B) = P(A) x P(B|A) P(A and B) = P(B) x P(A|B) Conditional Probability: The notation P(A, given B) denotes the probability that event A will occur given that event B has occurred. Insert Conditional Probability Rule How to Use the Multiplication Rules 1. First determine whether A and B are independent events. If P(A) = P(A|B), then the events are independent. 2. If A and B are independent events: P(A and B) = P(A) x P(B). 3. If A and B are any events, P(A and B) = P(A) x P(B|A) or P(A and B) = P(B) x P(A|B). What does Conditional Probability Tell Us? Conditional probability of two events A and B tell us: ● The probability that event A will happen under the assumption that event B has happened (or is guaranteed to happen in the future). This probability is designated P(A|B) and is read “probability of A given event B.” Note that P(A|B) might be larger or smaller than P(A). ● The probability that event B will happen under the assumption that event A has happened. This probability is designated P(B|A). Note that P(A|B) and P(B|A) are not necessarily equal. ● If P(A|B) = P(A) or P(B|A) = P(B), then events A and B are independent. This means the occurrence of one of the events does not change the probability that the other event will occur. ● Conditional probabilities enter into the calculations that two events A and B will both happen together. P(A and B) = P(A) x P(B|A) also P(A and B) = P(A)x P(B) In the case that events A and B are independent, then the formulas for P(A and B) simplify to. P(A and B) = P(A) x P(B). ● If we know the values of P(A and B) and P(B), then we can calculate the value of P(A|B). Mutually Exclusive/ Disjoint: Two events are mutually exclusive or disjoint if they cannot occur together. In particular, events A and B are mutually exclusive if P(A and B) = 0. Addition Rule for Mutually Exclusive Events A and B P(A or B) = P(A) + P(B) General Addition Rule for any Events A and B (Not Mutually Exclusive) P(A or B) = P(A) + P(B) P(A and B) How to Use the Addition Rules 1. First determine whether A and B are mutually exclusive events. If P(A and B) = 0, then the events are mutually exclusive. 2. If A and B are mutually exclusive events, P(A or B) = P(A) + P(B). 3. If A and B are any events, P(A or B) = P(A) + P(B) P(A and B). What Does the Fact that Two Events are Mutually Exclusive Tell Us? If two events A and B are mutually exclusive, then we know the occurrence of one of the events means that the other event will not happen. In terms of calculations, this tells us: ● P(A and B) = 0 for mutually exclusive events. ● P(A or B) = P(A) =P(B) for mutually exclusive events. ● P(A|B) =0 and P(B|A) = 0 for mutually exclusive events. That is, if event B occurs, then event A will not occur, and vice versa. 4.3 Tree and Counting Techniques ● The probability formula requires that we be able to determine the number of outcomes in the sample space. ● When an outcome of an experiment is composed of a series of events, the multiplication rule gives us the total number of outcomes . Tree Diagram: A visual display of the total number of outcomes of an experiment consisting of a series of events. Helps determine the total number of outcomes and individual outcomes. Factorial Notation: Procedure: What Do Counting Rules Tell Us? Counting rules tell us the total number of outcomes created by combining a sequence of events in specified ways. ● The multiplication rule tells us the total number of possible outcomes for a sequence of events. Tree diagrams provide a visual display of all the resulting outcomes. ● The permutation rule tells us the total number of ways we can arrange in order n distinct objects into a group of size r. ● The combination rule tells us how many ways we can form n distinct objects into a group of size r. The order of the objects is irrelevant.
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