Copyright © 2010 Pearson Education, Inc. Slide

Copyright © 2010 Pearson Education, Inc. Slide Solution: D

Copyright © 2010 Pearson Education, Inc. Chapter 14 From Randomness to Probability

Copyright © 2010 Pearson Education, Inc. Slide Dealing with Random Phenomena 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. Each occasion upon which we observe a random phenomenon is called a trial. Example: flipping a coin. At each trial, we note the value of the random phenomenon, and call it an outcome. Example: tails When we combine outcomes, the resulting combination is an event. Example: Flipping a coin twice and getting both tails. The collection of all possible outcomes is called the sample space. Example: List all the possible outcomes of flipping two coins.

Copyright © 2010 Pearson Education, Inc. Slide Independent Events - the outcome of one trial doesn’t influence or change the outcome of another. Example: Are the following independent events? a. coin flips b. Rolling a die or a pair of dice. c. Having an IPod and have an ITunes gift card. d. Your grade in AP statistics and your grade in trigonometry.

Copyright © 2010 Pearson Education, Inc. Slide The Law of Large Numbers The Law of Large Numbers (LLN) says that the long-run relative frequency of repeated independent events gets closer and closer to a single value. We call the single value the probability of the event. Because this definition is based on repeatedly observing the event’s outcome, this definition of probability is often called empirical probability.

Copyright © 2010 Pearson Education, Inc. Slide The Nonexistent Law of Averages The LLN says nothing about short-run behavior. Relative frequencies even out only in the long run, and this long run is really long (infinitely long, in fact). The so called Law of Averages (that an outcome of a random event that hasn’t occurred in many trials is “due” to occur) doesn’t exist at all. Example: One common proposal for beating the lottery is to note which numbers have come up lately, eliminate those from consideration and bet on the numbers that haven’t come up for a long time. Proponents of this method argue that in the long run, every number should be selected often, so those that haven’t come up are due. Explain why this is faulty reasoning.

Copyright © 2010 Pearson Education, Inc. Slide The probability of an event is the number of outcomes in the event divided by the total number of possible outcomes. P(A) = Modeling Probability # of outcomes in A # of possible outcomes

Copyright © 2010 Pearson Education, Inc. Slide Formal Probability 1.Two requirements for a probability: A probability is a number between 0 and 1. For any event A, 0 ≤ P(A) ≤ 1.

Copyright © 2010 Pearson Education, Inc. Slide Formal Probability (cont.) 2.Probability Assignment Rule: The probability of the set of all possible outcomes of a trial must be 1. P(S) = 1 (S represents the set of all possible outcomes.)

Copyright © 2010 Pearson Education, Inc. Slide Formal Probability (cont.) 3.Complement Rule:  The set of outcomes that are not in the event A is called the complement of A, denoted A C.  The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(A C )

Copyright © 2010 Pearson Education, Inc. Slide Example: When we arrive at the intersection of Altama and Community Rd. the probability the light is green is about 35% of the time. If P(green) =.35, what is the probability the light isn’t green when you get to Altama and Community?

Copyright © 2010 Pearson Education, Inc. Slide Formal Probability (cont.) 4.Addition Rule: Events that have no outcomes in common (and, thus, cannot occur together) are called disjoint (or mutually exclusive).

Copyright © 2010 Pearson Education, Inc. Slide Formal Probability (cont.) 4.Addition Rule (cont.): For two disjoint events A and B, the probability that one or the other occurs is the sum of the probabilities of the two events. P(A  B) = P(A) + P(B), provided that A and B are disjoint.

Copyright © 2010 Pearson Education, Inc. Slide Example: Knowing the P(green) =.35 and P(yellow) =.04, when we get to the light at the corner of Altama and Community: a. What is the probability the light is green or yellow? Written P(green yellow). b. When you get to the light are the events of the light being green and yellow disjoint or independent? c. What is the probability the light is red?

Copyright © 2010 Pearson Education, Inc. Slide Formal Probability (cont.) 5.Multiplication Rule: For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events. P(A  B) = P(A)  P(B), provided that A and B are independent.

Copyright © 2010 Pearson Education, Inc. Slide Example: Opinion polling organizations contact their respondents by telephone. In 1990s this method could reach about 60% of US households. By 2003, the contact rate had risen to 76%. We can reasonably assume each household’s response to be independent of the others. What is the probability that … a. The interviewer successfully contacts the next household on the list? b. The interviewer successfully contacts both of the next two households on the list? c. The interviewer’s first successful contact is the third household on the list? d. The interviewer makes at least one successful contact among the next five households on the list?

Copyright © 2010 Pearson Education, Inc. Slide Formal Probability (cont.) 5.Multiplication Rule (cont.): Two independent events A and B are not disjoint, provided the two events have probabilities greater than zero:

Copyright © 2010 Pearson Education, Inc. Slide Formal Probability (cont.) 5.Multiplication Rule: Many Statistics methods require an Independence Assumption, but assuming independence doesn’t make it true. Always Think about whether that assumption is reasonable before using the Multiplication Rule.

Copyright © 2010 Pearson Education, Inc. Slide Formal Probability - Notation In this text we use the notation P(A  B) and P(A  B). In other situations, you might see the following: P(A or B) instead of P(A  B) P(A and B) instead of P(A  B)

Copyright © 2010 Pearson Education, Inc. Slide Homework: Pg odd