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AP Statistics Chapter 6 - Probability

6.1: Simulations

SIMULATION
The imitation of chance behavior, based on a model that accurately reflects the experiment under consideration, is called a simulation.

THE STEPS OF A SIMULATION
1. State the problem or describe the experiment.
2. State the assumptions.
3. Assign digits to represent outcomes.
4. Simulate many repetitions.
5. State your conclusions.


6.2: Probability Models

RANDOMNESS AND PROBABILITY
We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number
of repetitions.

The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is long-term relative frequency.

PROBABILITY MODELS
The sample space S of a random phenomenon is the set of all possible outcomes.

An event is any outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.

A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of
assigning probabilities to events.

MULTIPLICATION PRINCIPLE
If you can do one task in a number of ways and a second task in b number
of ways, then both tasks can be done in a x b number of ways.

PROBABILITY RULES
Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.
Rule 2. If S is the sample space in a probability model, then P(S) = 1.
Rule 3. The complement of any event A is the event that A does not occur, written as . The complement rule states that
P() = 1 – P(A)
Rule 4. Two events A and B are disjoint (also called mutually exclusive) if they have no outcomes in common and so can never occur simultaneously. If A and B are disjoint, P(A or B) = P(A) + P(B). This is the addition rule for disjoint events.

THE MULTIPLICATION RULE FOR INDEPENDENT EVENTS
Rule 5. Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are
independent, P(A and B) = P(A)P(B). This is the multiplication rule for independent events.


6.3: General Probability Rules

ADDITION RULE FOR DISJOINT EVENTS
If events A, B, and C are disjoint in the sense that no two have any outcomes in common, then
P(one or more of A, B, C) = P(A) + P(B) + P(C). This rule extends to any number of disjoint events.

GENERAL ADDITION RULE FOR UNIONS OF TWO EVENTS
For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B)

GENERAL MULTIPLICATION RULE FOR ANY TWO EVENTS
The probability that both of two events A and B happen together can be found by

P(A and B) = P(A)P(B | A)


Here P(B | A) is the conditional probability that B occurs given that A has occured.

DEFINITION OF CONDITIONAL PROBABILITY
When P(A) > 0, the conditional probability of B given A is

INDEPENDENT EVENTS
Two events A and B that both have positive probability are independent if P(B | A) = P(B)

 

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