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Chapter 4, Part 1 Basic ideas of Probability Relative Frequency, Classical Probability Compound Events, The Addition Rule Disjoint Events 1 Idea of Probability • Probability is the science of “random” phenomena or “chance” behavior. • Many phenomena are unpredictable when observed only once, but follow a general pattern if observed many times. • Examples of “random” phenomena: – Coin flips, rolling dice, drawing cards – Drawing names/numbers out of a hat – ** Choosing a sample from population ** 2 Modeling Random Behavior • When we observe “random” behavior, we do not try to predict the results of a single observation. Instead, we… – Consider every possible result that could happen on a given observation. These are called outcomes for the random phenomenon. – Measure the “chance” that a given outcome will occur on a particular observation. This is called a probability. 3 Basic Terminology • An event is any set of outcomes or results of a given random phenomenon. • An outcome (or simple event) is a single result that cannot be broken down into simpler components. • The Sample Space for a random phenomenon is the set of all possible outcomes. 4 Basic Terminology, Examples • I randomly select one student from my class list. • Some (non-simple) events: – – – – The student is in Row 1 The student is absent The student is female The student’s Exam 1 score was above 90%. • Each individual student on the class list is an outcome (simple event). • The Sample Space is the set of all students on my class list. 5 Probability • When we try to model/describe a random phenomenon, each event is assigned a number, called the probability of the event. • Probability measures how likely it is that an event will occur. Events with higher probability are more likely to happen (or tend to happen more frequently) • P(A) denotes the probability of event A. We always require that 0 ≤ P(A) ≤ 1. Possible Values for Probabilities: 0 ≤ P(A) ≤ 1 Three Views of Probability • Relative Frequency (Empirical Probability): – Actual Data is used to estimate the probability of various events. • Classical Approach (Theoretical Probability): – Assign probabilities in a way that satisfies a set of formal mathematical rules. • Subjective Probability (“Expert Opinion”): – Use prior knowledge from a similar situation in order to estimate probabilities. Relative Frequency • Estimate probability from actual data. • Take many observations of a random phenomenon, and count how many times a particular event occurs. • The relative frequency of the event is: (# of occurrences) / (# of observations). • In other words, for what proportion of observations did the event occur? • It may be helpful for you to think of this as a percentage. 9 Relative Frequency: Examples • Earlier in the class, you drew a slip of paper from the hat. This gives us about 30 observations of a random phenomenon. • Using our actual data, we now compute: • Relative frequency of “Blue” = • Relative frequency of “Pink” = Classical Probability • Our random phenomenon has Sample Space with finitely many outcomes, say n of them. • Assume that each outcome is equally likely to occur on any given observation. • The (classical) probability of the event is: (# of outcomes in the event)/(total # of outcomes) • NOTE: This is actually just a special case of a more general approach (theoretical probability). Classical Probability, Examples • I randomly select one student from class (among those currently present). • Assume that each student (outcome) is equally likely. • Compute the (classical) probability of: – The student is in Row 1. – The student is in the back row. – The student is texting on his/her cell phone. Why Classical Probability? • Suppose our random phenomenon is “Choose a sample of N individuals from a large population.” • Each outcome is a group of N individuals. • Note that N is NOT the total number of outcomes (that number is MUCH BIGGER) • If we assume that “all outcomes are equally likely,” then we are talking about…? The Law of Large Numbers • Relative Frequency: Estimate probability using actual observational data. • Classical Probability: Compute using knowledge of the Sample Space. • Question: What if our knowledge of the Sample Space is incomplete? – Example: We know the hat has only pink and blue slips, but we don’t know how many of each kind. 14 The Law of Large Numbers • Under a certain condition (independent observations, discussed next time): As we increase the number of observations, the Relative Frequency of an event tends to be closer to the (theoretical) Probability of that event. • Relative Frequency estimates Probability. With more observations, you are more likely to get a better estimate. 15 Compound Events • Let A, B be two events. For example, if I choose a student from those in class: – A = “The student I choose is male.” – B = “The student I choose is in the back row.” • Each event is actually a set of outcomes, but you can think of each event as some kind of condition/requirement. • The event “A or B” is the set of outcomes that meet at least one (but possibly both) of the given requirements. Compound Events • The event “A and B” is the set of outcomes that meet both of the given requirements. • In the previous example: – “A or B”: The chosen student is male, in the back row, or both (at least one condition is met). – “A and B”: The chosen student is male AND in the back row (both conditions are met). The Addition Rule • If A and B are any events, then we have: P(A or B) = P(A) + P(B) – P(A and B) • Here’s a version that’s useful when “all outcomes are equally likely”: Let #(A) be the number of outcomes in event A. Then #(A or B) = #(A) + #(B) - #(A and B) 18 Disjoint Events • If events A and B have no outcomes in common (they cannot occur at the same time), then we say that they are disjoint. – Example: “Student is in Row 1” and “Student is in Row 3” • In this case P(A and B) = 0. So the Addition Rule becomes: P(A or B) = P(A) + P(B) Complementary Events • Given an event A, the complement of A is the set of outcomes that are not in A. – Notation: A is the complement of A. • Example: Choose a student from class: – A = “Student is in Row 1” – B = “Student is Female” • Note that an event and its complement will always be disjoint. Formulas for Complementary Events P( A) P( A) 1 P( A) 1 P( A) P( A) 1 P( A)