Statistics Chapter 4
Probability of two events A and B
This is the intersection of A and B is he event consisting of the sample space outcomes belonging to both A and B. The intersection is denoted by A⋂B. The probability is denoted P(A⋂B) which is the probability that both A or B or Both will occur.
⋂
This means the intersection. Used to denote that the sample space contains both.
Mutually exclusive events
Two events A and B are mutually exclusive if they have no sample space outcomes in common. So P(A⋂B) = 0
independent events
When the probability of one event is not influenced by whether another event occurs, the event are said to be independent.
Probability
A number that measures the chance or likelihood that an event will occur when an experiment is carried out.
Experiment
A process of observation that has an uncertain outcome.
Event
A set of sample space outcomes.
Bayes Theorem
A theorem that is used to computer posterior probabilities by revising prior probabilities.
Bayesian Statistics
An area of statistics that uses Bayes Theorem to update prior belief about a probabilitiy or population parameter to posterior belief.
Notation for probability of an event
P(E) Probability (P) of event (E) happening.
A⋃B
Asking the whether event A or B OR BOTH will occur.
A⋂B
Asking whether both event A and B will occur. The sample space is the intersection of both A and B.
Calculate the probability of an event given all outcomes are equally likely
Equal to the ratio P = # sample space outcomes that correspond to the event / total number of sample space out comes.
mutually exclusive events
Events that have no sample space outcomes in common and therefore cannot occur simultanously
Multiplication Rule for independent events
If A and B are independent events, then P(A⋂B...⋂N) = P(A)*P(B)...*P(N)
Complement (of an event)
If A is an event, the complement of A is the event that A will not occur
Denoting the compliment of an event
If the probability of an event happening is P(A), then the probability of an event not happening is P(A-Hat).
Two events are independent if and only if
P(A|B) = P(A) OR P(A|B) = P(B)
Conditional probability that A will occur given that B will occur.
P(A|B) = P(A⋂B)/P(B) This equation can be rearranged various ways ie P(A⋂B)=P(B)*P(A|B) OR P(A⋂B)=P(A)*P(A|B)
The Addition Rule
P(A⋃B) = P(A) + P(B) - P(A⋂B). You subtract P(A⋂B) because you will count the overlap between A and B twice and get an incorrect probability.
Addition rule for mutually exclusive events
P(A⋃B...N) = P(A) + P(B) + P(N) There is no overlap - so no need to subtract out P(A⋂B).
What is the probability of an event?
The probability of an event is the sum of the probabilities of the sample space outcomes that correspond to the event.
Rules for probabilities
The probability of each experimental outcome must be between 0 and 1. The sum of all experimental outcomes must sum to 1
Conditional Probability
The probability of the event A, given the condition that the event B has occurred, is written as P(A|B) - pronounced the probability of A given B.
conditional probabilitiy
The probability that one event will occur given that we know that another event occurs. (pg 171)
Decision theory
an approach that helps decision makers to make intelligent choices.
prior probabilitiy
the initial probability that an event will occur.
Dependent event
when the probaiblity of one event is influenced by whether another event occurs. The events are said to be dependent.