# Probability

by Jane_l Yang
# 1. Random Circumstance

## 1.1. outcome not determined until we see it

## 1.2. Outcome already determined but our knowledge is uncertain

# 2. Relative Frequency Interpretation

## 2.1. Make assumption about Physical world

## 2.2. Make direct observation of how often something happens over many, many repetitions of the situation.

# 3. Number between 0 and 1

## 3.1. The possible outcome of a random circumstance

## 3.2. Meets 1st condition for valid probablity

# 4. Personal Probability (subjective)

## 4.1. Degree an individual believes it will happen

## 4.2. Coherent (no contradiction)

# 5. Uses percentages based on proportions of individuals

# 6. Definitions

## 6.1. Sample Space

### 6.1.1. With Replacement

### 6.1.2. Without Replacement

## 6.2. Simple Event

### 6.2.1. Sum of all Simple Events = 1

### 6.2.2. If equally likely then probability = 1/k

## 6.3. Event

# 7. Events

## 7.1. Complementary ("Opposite")

## 7.2. mutually exclusive (disjoint)

## 7.3. Independent

## 7.4. Dependent

## 7.5. Conditional Probablity

### 7.5.1. P(B|A) = probability of B given A

# 8. Rules for Finding Probabilities

## 8.1. Rule 1 (Not the event)

### 8.1.1. Probability an event does not occur

8.1.1.1. P(A)+P(A^c)=1

## 8.2. Rule 2 (Addition)

### 8.2.1. Probability that either of two events happens

8.2.1.1. Rule 2a (general)

8.2.1.1.1. P(A or B)=P(A)+P(B)-P(A and B)

8.2.1.2. Rule 2b (mutually exclusive events)

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

## 8.3. Rule 3 (Multiplication)

### 8.3.1. Probability that two or more events occur together

8.3.1.1. Rule 3a (general)

8.3.1.1.1. P(A and B)=P(A)PB|A)=P(B)PA|B)

8.3.1.2. Rule 3b (independent events)

8.3.1.2.1. PA and B)=P(A)P(B)

8.3.1.3. Rule 3b (extension)

8.3.1.3.1. P(A1 and A2...and An)=P(A1)P(A2)...(PAn)

## 8.4. Rule 4 (Conditional Probability)

### 8.4.1. Determining a conditional probability

8.4.1.1. P(B|A)=P(A andB)/P(A)

# 9. Philosophical Issue

## 9.1. Outcome has been determined but is still unknown

## 9.2. Confidence vs Probability

# 10. Strategies

## 10.1. Bayes Rule

### 10.1.1. P(A|B)=P(A and B)/PB|A)P(A)+P(B|A^c)P(A^c)

## 10.2. Two-way Tables: Hypothetical 100,000

## 10.3. Tree Diagram