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Chapter 7 Probability
by Jenn Handler
# Chapter 7 Probability

## 7.1 Random Circumstances

### Random Circumstance

### Probability

## 7.2 Interptetations of
Probability

### Relative Frequency

### Personal Probability

### Subjective Probability

### Coherent Probabilities

### Determining the Relative
Frequency Probability of an
outcome

## 7.3 Probability Definitions
and Relationships

### Simple Event

### Sample Space

### Event

### Complimentary events

### Independent Events

### Dependent Events

### Mutually Exclusive (Disjoint) Events

### Complement

### Assigning probabilities to
simple events

### Two conditions for valid probabilities

### Conditional Probabilities

## 7.4 Basic Rules for Finding
Probabilities

### Probability Rule 1: Complements

### Probability Rule 2: Addition

### Probability Rule 3: Multiplication

### Probability Rule 4: Conditional
Probability

### Sample without Replacement

### Sample with Replacement

## 7.5 Stragegies for Finding
Complicated Probabilities

### Bayes Rule

### Tree Diagrams

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When the outcome is unpredictable, the outcome is not determined until we observe it

How likely it is that a particular outcome will be the result of a random circumstance. The total of the assigned probablities must equal 1

For situations that we can imagine repeating many times, we define the probability of a specific outcome as the proportion of times it would occur over the long run (numbers between 0 and 1)

The degree to which a given individual believes that the event will happen

Another term used for personal probability because the degree of belief may be different for each individual

That your personal probability of one event doesn't contradict your personal probability of another, EXAMPLE If you believe the probability of getting an A in Statistics is .4 then you must also believe the probability of not getting an A is .6 if they are to be coherent probabilities.

Making an assumption about the physical world, EXAMPLE The probability of a flipped coin showing heads up is 1/2

Making a direct observation of how ofter something happens, EXAMPLE (according to the book) The probability that a live birth will result in a male is about .5117

One outcome in the sample space, a possible outcome of a random circumstance

The collection of unique overlapping possible outcomes of a random circumstance

A collection of one or more simple events in the sample space(Events are often written using capital letters A,B,C, & ect.)

Opposite events EXAMPLE you have blonde hair or you don't have blonde hair

If knowing that one will occur or has occurred does not change the probability that other will occur

If knowing that one will occur or has occurred changes the probability that the other occurs

Do not contain any of the same simple events (outcomes)

When two events do not contain any of the same simple events and together they cover the entire sample space (For an event A, the notation A^c represents the complement of A), Probabilities of Compliment events must sum to 1. P(A) + P(A^c) = 1 so P(A^c) = 1 - P(A)

P(A) is used to denote the probability that event A occurs

Each probability is between 0 and 1

The sum of the probabilities over all possible simple events is 1

P(B), the unconditional probability that the event B occurs

P(BlA), probability of B given A, the conditional probability that the event B occurs given that we know A has occured or will occur

The long-run relative frequency with which event B occurs when circumstances are such that A also occurs

(for "not the event") To find the probability of A^c, the complement of A, use P(A^C) = 1 - P(A). P(A^c) is the probability that the event will not occur

Rule 2a, (General) P(A+B) = P(A) + P(B) - P(A and B)

Rule 2b, (Mutually Exclusive Events) If A and B are mutually exclusive events, P(A or B) = P(A) + P(B)

To find the probability that two events A and B both occur simultaneously or in a sequence

Rule 3a, P(A and B) = P(A)*P(B given A) or P(A and B) = P(B)*P(A given B)

Rule 3b, For INDEPENDENT events, P(A and B) = P(A)*P(B)

P(B given A) = P(A and B)/P(A) or P(A given B) = P(B and A)/P(B)

When individuals are not eligible for subsequent selection

When individuals are returned to the eligible pool for each selection

P(A given B) = P(A and B)/[P(B given A)*P(A) + P(B given A^c)*P(A)}

A schematic representation of the sequence of events and their probabilities, including conditional probabilities based on previous events for the events that happen sequentially

Steps for creating a tree diagram, Determine the first random circumstance in the sequence, and create the first set of branches to illustrate possible outcomes for it. Create one branch for each outcome, and write the associated probability on the branch, Determine the next random circumstance, and append the branches for the possible outcomes to each of the branches in step 1., Continue this process for as many steps as necassary, To determine the probability of any particular sequence of branches, multiply the probabilities on those branches, To determine the probability of any collection of sequences of branches, add the individual probabilities for those sequences