# Probability

by Chelsea Gray
# 1. Random Circumstances

## 1.1. Random circumstance: one in which the outcome is unpredictable. In many cases the outcome is not determined until we observe it.

## 1.2. Probability: a number between 0 and 1 that is assigned to a possible outcome of a random circumstance.

# 2. Interpretations of Probability

## 2.1. Relative frequency: In situations that we can imagine repeating many times, we define the probability of a specific out come as the proportion of times it would occur over the long run.

## 2.2. Personal probability: the degree to which a given individual believes that the event will happen

## 2.3. Subjective probability: The degree of belief may be different for each individual

## 2.4. Coherent: your personal probability of one event doesn't contradict your personal probability of another

# 3. Probability Definitions and Relationships

## 3.1. Sample space: the collection of unique, non-overlapping possible outcomes of a random circumstance.

### 3.1.1. Simple Event: one outcome in the sample space. Simple event is a possible outcome of a random circumstance.

3.1.1.1. Event: a collection of one or more simple events in the sample space. Often written using capital letters.

## 3.2. Complementary event: the two events do not contain any of the same simple events and together they cover the entire sample space.

## 3.3. Mutually exclusive/disjoint: the events do not contain any of the same simple events (outcomes).

## 3.4. Independent event: knowing that one will occur (or has occurred) does not change the probability that the other occurs.

## 3.5. Dependent event: knowing that one will occur (or has occurred) changes the probability that the other occurs.

## 3.6. Conditional probability: of the event B, given that the event A occurs, is the long-run relative frequency with which event B occurs when circumstances are such that A also occurs.

# 4. Basic Rules for Finding Probabilities

## 4.1. Rule 1 ( "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 event A won't occur

## 4.2. Rule 2 (addition rule for "either/or"): to find the probability that either A or B or both happen.

### 4.2.1. Rule 2a (general): P(A or B) = P(A and B)

### 4.2.2. Rule 2b (for mutually exclusive events): if A and B are mutually exclusive events, P(A or B) = P(A) + P(B)

## 4.3. Rule 3 (multiplication rule for "and"): to find the probability that two events A and B both occur simultaneously or in a sequence.

### 4.3.1. Rule 3a (general): P(A and B) = P(A)P(B|A) = P(B)P(A|B)

### 4.3.2. Rule 3b (for independent events): if A and B are independent events, P(A and B) = P(A)P(B)

## 4.4. Rule 4 (conditional probability): an algebraic restatement of rule 3a

### 4.4.1. P(B|A) = P(A and B)/P(A)

## 4.5. Sample with replacement: individuals are returned to the eligible pool for each selection.

## 4.6. Sample without replacement: individuals are not eligible for subsequent selection.

# 5. Strategies for Finding Complicated Probabilities

## 5.1. Bayes Rule: P(A|B) = P(A and B)/P(B|A)P(A) + P(B|A^c)P(A^c)

## 5.2. Tree diagram: a schematic representation of the sequence of events and their probabilities, including conditional probabilities based on previous events that happen sequentially.