Chapter 6 Stats
Requirements of Probabilites
1) The probability of any outcome must lie between 0 and 1, inclusive for each i. 2) The sum of the probabilities assigned to all outcomes in the sample space must add up to 1.
THREE RULES
1. Complement Rule, 2. Multiplication Rule 3. Addition Rule
Several types of combinations and relationships between events
1. Complement of an event 2. Intersection of two events 3. Union of two events 4. Mutually exclusive events 5. Dependent events 6. Independent events
Intersection of two events
A and B consists of all points that are in both A and B. That is, the intersection of A and B occurs when both A and B occur. It is denoted: A and B
Union of two events
A and B consists of all points that are in either A or B or both. That is, the union of A and B occurs when either A or B or both occur. The union of A and B is denoted: A or B
Simple Event
An individual outcome of the sample space
3 Approaches to assigning probabilities P(Oi), to an outcome, Oi.
Classical Approach, Relative Frequency Approach, Subjective Approach
Marginal Probabilities
Computed by adding across rows or down columns; are so named because they are calculated in the margins of the table. Probability of only one outcome.
The Subjective Approach (When it is not reasonable to use the classical approach and there is no history of the outcomes)
Defines probability as the degree of belief that we hold in the occurrence of an event
Relative frequency approach
Defines probability as the long-run relative frequency with which an outcome occurs.
Classical Approach
If an experiment has n possible outcomes, the classical approach would force us to assign a probability of 1/n to each outcome. It is necessary to determine the number of possible outcomes (n) for the experiment.
Out comes of Experiment must be Mutually Exclusive
No two outcomes can occur at the same time. When they don't have anything in common.
Addition Rule
The addition rule is used to compute the probability of event A or B or both A and B occurring; i.e. the union of A and B. P(A or B) = P(A) + P(B) - P(A and B) P(A or B) = P(A) + P(B) - P(A and B) ^^^^If A and B are mutually exclusive, then this term goes to zero
Joint Probability
The probability of the intersection; namely P(A and B). If there is not joint probability it will equal= 0.
Dependent
Two events A and B are said to be DEPENDENT if the probability of occurrence of one event (A, say) is affected by the occurrence of the other event (B).
Independent
Two events are said to be independent if the probability of occurrence of one event is not affected by the occurrence of the other event. More specifically, two events A and B are said to be independent if P(A|B) = P(A) or P(B|A) = P(B) The outcome of one does not affect the outcome of another.
Mutually Exclusive
Two events are said to be mutually exclusive if they cannot occur together. In this case, their joint probability is 0. The events A and B are mutually exclusive since they have no points in common, so they cannot occur at the same time. P(A and B) = 0
Conditional Probability
Used to determine how two events are related. That is, we can determine the probability of occurrence of one event given that another related event has already occurred. For two events A and B, conditional probabilities are written as P(A | B) or P(B | A). **Given that one event has already ocurred, the probability that another event will occur. Can be written in terms of joint probabilities and marginal probabilities.
Union
We stated earlier that the union of two events A and B is denoted as A or B. It occurs when either A or B or both occur.
Sample Space of an Experiment
a list of all possible outcomes of the experiment
Event
a set of one or more outcomes (simple events) considered as a group.
Random Experiment
an action or process that leads to one of several possible outcomes (Oi).
Complement of event A
defined to be the event consisting of all points that are "not in A". That is, the complement of event A is the event that occurs when event A does not occur. P(Ac ) = 1- P(A) for any event A
Outcomes of Experiment must be exhaustive
means that all possible outcomes must be included
Complement Rule
the complement rule gives us the probability of an event not occurring. That is: P(AC) = 1 - P(A)
Multiplication Rule
used to calculate the joint probability of two events. It is based on the formula for conditional probability defined earlier: P(A/B)=P(A and B) P(B) If we cross multiply we have: P(A and B) = P(A | B)•P(B) Like wise: P(A and B) = P(B | A) • P(A) In addition, if A and B are independent events, then: P(A and B) = P(A)•P(B)
