ch12 (Probability Part II - general rules) - York · PDF filech12 (Probability Part II - general rules) ch12-links . Kerns probability using R Ch12 image. ch12 exercise: 12.1, 12.3,

ch12 (Probability Part II - general rules)

ch12-links Kerns probability using R Ch12 image ch12 exercise: 12.1, 12.3, 12.4, 12.6, 12.14, 12.38

Venn Diagrams

Further Rules of Probability

Conditional Probability

The Binomial Distributions (chapter 13) An experiment is a repeatable procedure or process for making an observation. An outcome is a possible observation resulting from an experiment. The sample space (S) of the experiment is the set of all possible outcomes. An event (E) is a subset of the sample space (that is, a set of outcomes).

There are four rules or axioms that all probabilities should follow. The first rule or axiom states that all probabilities lie between 0 and 1, i.e. the probability of an event E in the sample space is 0≤ P(E) ≤1 .

The second rule or axiom states that the sample space S is exhaustive - some outcome must occur: P(S) = 1.

Two events A and B are disjoint if they have no outcomes in common; disjoint events cannot occur simultaneously. The third rule is that the probability of occurrence of one or the other of two disjoint events is the sum of their separate probabilities of occurrence: For A and B disjoint, P(A or B) = P(A) + P(B). The fourth rule states that the probability that an event E does not occur is the complement of its probability of occurrence: P(not E) = 1 - P(E).

CHAPTER 12: General Rules of Probability

Lecture PowerPoint Slides

The Basic Practice of Statistics6th Edition

Moore / Notz / Fligner

Chapter 12 Concepts2

Independence and the Multiplication Rule

The General Addition Rule

Conditional Probability

The General Multiplication Rule

Tree Diagrams

Chapter 12 Objectives3

Define independent events

Determine whether two events are independent

Apply the general addition rule

Define conditional probability

Compute conditional probabilities

Apply the general multiplication rule

Describe chance behavior with a tree diagram

Everything in this chapter follows from the four rules we learned in Chapter 10:

Probability Rules

Rule 1. The probability P(A) of any event A satisfies 0 ≤

P(A) ≤

1.

Rule 2. If S is the sample space in a probability model, then P(S) = 1.

Rule 3. If A and B are disjoint, P(A or B) = P(A) + P(B).

This is the addition rule for disjoint events.

Rule 4. For any event A, P(A does not occur) = 1 – P(A).

Rule 1. The probability P(A) of any event A satisfies 0 ≤

P(A) ≤

1.

Rule 2. If S is the sample space in a probability model, then P(S) = 1.

Rule 3. If A and B are disjoint, P(A or B) = P(A) + P(B).

This is the addition rule for disjoint events.

Rule 4. For any event A, P(A does not occur) = 1 – P(A).

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Venn DiagramsSometimes it is helpful to draw a picture to display relations among several events. A picture that shows the sample space S as a rectangular area and events as areas within S is called a Venn diagram.

Two disjoint events:

Two events that are not disjoint, and the event {A and B} consisting of the outcomes they have in common:

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Multiplication Rule for Independent Events

If two events A and B do not influence each other, and if knowledge about one does not change the probability of the other, the events are said to be independent of each other.

Multiplication Rule for Independent EventsTwo events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent:

P(A and B) = P(A)

P(B)

Multiplication Rule for Independent EventsTwo events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent:

P(A and B) = P(A)

P(B)

The General Addition Rule7

We know if A and B are disjoint events,

P(A or B) = P(A) + P(B)

Addition Rule for Any Two EventsFor any two events A and B:

P(A or B) = P(A) + P(B) – P(A and B)

Addition Rule for Any Two EventsFor any two events A and B:

P(A or B) = P(A) + P(B) – P(A and B)

Conditional Probability

The probability we assign to an event can change if we know that some other event has occurred. This idea is the key to many applications of probability.

When we are trying to find the probability that one event will happen under the condition that some other event is already known to have occurred, we are trying to determine a conditional probability.

The probability that one event happens given that another event is already known to have happened is called a conditional probability.

When P(A) > 0, the probability that event B happens given that event A has happened is found by:

The probability that one event happens given that another event is already known to have happened is called a conditional probability.

When P(A) > 0, the probability that event B happens given that event A has happened is found by:

P(B | A) P(A and B)P(A)

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The General Multiplication Rule9

The probability that events A and B both occur can be found using the general multiplication rule

P(A and B) = P(A) • P(B | A)

where P(B | A) is the conditional probability that event B occurs given that event A has already occurred.

The probability that events A and B both occur can be found using the general multiplication rule

P(A and B) = P(A) • P(B | A)

where P(B | A) is the conditional probability that event B occurs given that event A has already occurred.

The definition of conditional probability reminds us that in principle all probabilities, including conditional probabilities, can be found from the assignment of probabilities to events that describe a random phenomenon. The definition of conditional probability then turns into a rule for finding the probability that both of two events occur.

Note: Two events A and B that both have positive probability are independent if:

P(B|A) = P(B)

Note: Two events A and B that both have positive probability are independent if:

P(B|A) = P(B)

Suppose two cards are dealt from a well-shuffled deck. They are placed side-by-side face down on the table, the first card to the left of the second. a) What is the probability that the first card (A1) is an ace? a) P(A1) = 4/52 = 1/13 b) What is the probability that the second card (A2) is an ace? b) (A1 and A2) or (not A1 and A2) P(A1 and A2) +P(not A1 and A2) = 1/13 (4/52*3/51)+(48/52*4/51) = (1/52*51)*(4*3+48*4) = 204/52*51 = 4/52 = 1/13 c) What is the probability that both cards are aces? {P(A1 and A2)} c) P(A1 and A2) = P(A2/A1) P(A1) = (3/51) * (4/52) = (1/17)*(1/13) = 1/221 d) What is the probability that at least one of the cards is an ace? {P(A1 or A2)} d) P(A1) + P(A2) – P(A1 and A2) = 1/13 + 1/13 – (4/52)*(3/51) = 2/13 – (1/13)*(1/17) = 34/221 - 1/221 = 33/221 e) If the first card is turned over and seen to be an ace, what is now the probability that the second card is an ace? {P(A2 / A1)} e) [P(A2/A1)] = P(A2 and A1)/P(A1) = (4/52)* (3/51) / (4/52) = 1/17

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Tree Diagrams

We learned how to describe the sample space S of a chance process in Chapter 10. Another way to model chance behavior that involves a sequence of outcomes is to construct a tree diagram.

Consider flipping a coin twice.

What is the probability of getting two heads?

Sample Space:HH HT TH TT

So, P(two heads) = P(HH) = 1/4

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Example

The Pew Internet and American Life Project finds that 93% of teenagers (ages 12 to 17) use the Internet, and that 55% of online teens have posted a profile on a social-networking site.What percent of teens are online and have posted a profile?

P(online) 0.93P(profile | online) 0.55

P(online and have profile) P(online) P(profile | online)

(0.93)(0.55) 0.5115

51.15% of teens are online and have posted a profile.

Chapter 12 Objectives Review12

Define independent events

Determine whether two events are independent

Apply the general addition rule

Define conditional probability

Compute conditional probabilities

Apply the general multiplication rule

Describe chance behavior with a tree diagram