Chapter 4 Vocab

Random process

generates outcomes that are determined purely by chance

Probability

a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of trials

Law of Large Numbers

if we observe more and more trials of any random process, the proportion of times that specific outcome occurs approaches its probability

Simulation 

Imitates a random process in such a way that simulated outcomes are consistent with real-world outcomes

Probability model

a description of a random process that consists of two parts: a list of all possible outcomes and the probability of each outcome

Sample space

A list of all possible outcomes

Event

a subset of the possible outcomes from the sample space of a random process

P(A)

Probability equation, # of outcomes in event A/total number of outcomes in sample space

Basic Probability Rules

• The probability of any event is a number between 0 and 1

• All possible outcomes together must have probabilities that add up to 1

• The probability that an event does not occur is 1 minus the probability that the event does occur.

Complement Rule

P(A^C)=1 - P(A), where A^C is the complement event of A

Complement

the event that A does not occur

Mutually Excusive

If two events, A and B, have no outcomes in common and so can never occur together - that is, if P(A and B) = 0

Disjoint 

Synonym with mutually exclusive

Additional Rule for Mutually Exclusive Events

P(A and B) = P(A) + P(B)

General Addition Rule

P(A or B) = P(A) + P(B) - P(A and B) if A and B are any two events resulting from the same random process

Venn diagram

consists of one or more circles surrounded by a rectangle. Each circle represents an event. The region inside the rectangle represents the sample space of the random process.

Intersection 

The event “A and B” of events A and B. It consists of all outcomes that are common to both events, and is denoted by A n B

Union

The event “A or B” of events A and B. It consists of all outcomes that are in event A, event B, or both, and is denoted A U B

Conditional probability

The probability that one event happens given that another event is known to have happened.

P(A|B)

Conditional probability denotation

Conditional Probability Formula 

P(A|B) = P(A and B)/P(B) = P(AnB)/P(B) = P(bothe events occur)/P(given event occurs

Independent events

A and B is knowinf whether or not one event has occurred does not change the probability that the other event will happen. in other words, events A and B are independent if P(A|B) = P(A|B^C) = P(A) Alternatively, events A and B are independent if P(B|A) = P(B|A^C) = P(B)

General Multiplication Rule

For any random process, the probability that events A and B both occur: P(A and B) = P(AnB) = P(A) x P(B|A)

tree diagram

shows the sample space of a random process involving multiple stages. The probability of each outcome is shown on the corresponding branch of the tree. All the probabilities after the first stage are conditional probabilities.

Multiplication Rule for Independent Events

Events A and B are independent if and only if P(AnB) = P(A) x P(B)