Copyright © 2010 Pearson Education, Inc. Slide 14 - 1.
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Transcript of Copyright © 2010 Pearson Education, Inc. Slide 14 - 1.
Slide 14 - 4Copyright © 2010 Pearson Education, Inc.
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.
Slide 14 - 5Copyright © 2010 Pearson Education, Inc.
Independent Events - the outcome of one trial doesn’t influence or change the outcome of another.
Example: Are the following independent events?a. coin flipsb. 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.
Slide 14 - 6Copyright © 2010 Pearson Education, Inc.
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.
Slide 14 - 7Copyright © 2010 Pearson Education, Inc.
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.
Slide 14 - 8Copyright © 2010 Pearson Education, Inc.
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
Slide 14 - 9Copyright © 2010 Pearson Education, Inc.
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.
Slide 14 - 10Copyright © 2010 Pearson Education, Inc.
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.)
Slide 14 - 11Copyright © 2010 Pearson Education, Inc.
Formal Probability (cont.)
3. Complement Rule: The set of outcomes that are not in the event
A is called the complement of A, denoted AC. The probability of an event occurring is 1
minus the probability that it doesn’t occur: P(A) = 1 – P(AC)
Slide 14 - 12Copyright © 2010 Pearson Education, Inc.
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?
Slide 14 - 13Copyright © 2010 Pearson Education, Inc.
Formal Probability (cont.)
4. Addition Rule: Events that have no outcomes in common
(and, thus, cannot occur together) are called disjoint (or mutually exclusive).
Slide 14 - 14Copyright © 2010 Pearson Education, Inc.
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.
Slide 14 - 15Copyright © 2010 Pearson Education, Inc.
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?
Slide 14 - 16Copyright © 2010 Pearson Education, Inc.
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.
Slide 14 - 17Copyright © 2010 Pearson Education, Inc.
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?
Slide 14 - 18Copyright © 2010 Pearson Education, Inc.
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:
Slide 14 - 19Copyright © 2010 Pearson Education, Inc.
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.
Slide 14 - 20Copyright © 2010 Pearson Education, Inc.
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)