Probability
A number between 0 and 1 that describes the proportion of times any outcome of a chance process would occur in a very long series of repetitions.
Law of Large Numbers
If we observe more and more repetitions of any chance process, the proportion of times a specific outcome will occur approaches a single value (in the long run). In the short run, it is unpredictable.
Simulation
An imitation of chance behavior based on a model that accurately reflects the situation.
Performing a Simulation
1. State: Ask a question of interest.
2. Plan: Describe how to imitate the chance process, using a device such as cards, a random number generator, or a table of random digits.
3. Do: Perform many repetitions.
4. Conclude: Answer your question of interest
Sample Space (S)
The set of all possible outcomes of a chance process.
Probability Model
A description of some chance process that consists of two parts: a sample space S and a probability for each outcome.
Event
Any collection of outcomes from some chance process. A subset of the sample space. Usually designated by capital letters.
Basic Rules of Probability
1. The probability of any event is between 0 and 1.
2. All possible outcomes must add up to 1.
3. The probability of event does not occur is one minus the probability it does.
4. If two events have no outcomes in common, the probability one or the other o
Complement
Everything other than an outcome/event in the sample space.
Complement rule: P(A^c) = 1 - P(A)
Mutually Exclusive (Disjoint)
Two outcomes that have no outcomes in common so can never occur together. Can never be independent, because one can't happen with the other.
Venn Diagram
A way to illustrate the sample space of a chance process including two events, consisting of two circles representing the events.
Intersection (?)
All the outcomes in common between two events compared. P(A and B)
Union (U)
All the outcomes in the two events included. P(A or B)
General Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
Fixes the double counting problem because of the overlapping outcomes.
Conditional Probability
The probability an event will occur given another event has already occurred. Denoted by P(B|A). For example, the probability the person is a man given he is 30.
P(B|A) = P(A?B)/P(B)
P(A|B) = P(B?A)/P(A)
Independent Events
Two events in which the occurrence of one event does not change the probability the other with happen. P(A|B) = P(A), and P(B|A) = P(B)
General Multiplication Rule
Finds the probability both A and B occur using the formula:
P(A and B) = P(A?B) = P(A) * P(B|A)
Tree Diagram
Displays the sample space of a process involving a sequence of events, with each each subsequent event branching out from the first
Multiplication Rule for Independent Events
If A and B are independent, probability A and B both occur is:
P(A?B) = P(A) * P(B)
Finding if Two Events are Independent
1. P(B|A) = P(B)
2. P(A|B) = P(A)
Long-run behavior
probability approaches outcome
relative frequency
percent, in long-run is probability
addition rule for mutually exclusive events
P(A or B) = P(A) + P(B)
disjoint
P(A and B)=0
two-way table
table of counts that organizes data about two categorical variables
How to find the probability of at least one
P(at least one)=1-P(none)