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Probability for equally likely outcomes (f/N rule)
 to measure probability or to measure the possible outcomes (measuring uncertainty)
 Suppose an experiment has N possible outcomes, all equally likely. An event that can occur in f ways has probabilyt of f/N occurring:
 Probability of an event = f < number of ways event can occur
 N < total number of possible outcomes
 experiment: an action whose outcome cannot be predicted with curtainty
 event: some specified result that may or may not occur when an experiment is performed
 ex.
 f =7534
 N 75617 = 0.100
 Interpretation 10.0% of familied makte between so and so

Frequentist interpretation of probability( the meaning of probability)
 (when outcomes are equally likely probabilities are nothing more than percentage (relative frequency))
 When number of tosses is small the probability is flactuates a lot. When number of tosses is largethe probability stabelizes (50/50)
 Interpretation of probability: a probability near 0 (ex. 0.2) indicates that the event in question is very unlikely to occur when the experiment is performed.
 When a probability near 1 (100%)(0.8) suggests that the event is quite likely to occur.

Probability model
 although the frequentist interpretation is helpful for understanding the meaning of probability, it cannot be used as a definition of probability. One common way to define probabilities is to specify a probability model  a mathematical description of the experiment based on certain primary aspects and assumptions.
 Equallikelihood model: is a axample of probability model. Its primary aspect and assumption are that all possible outcomes are equally likely to occur.

Basic properties of probabilities
 Property 1: The probability of an event is always between 0 and 1, inclusive.
 Property 2: The probability of an event that cannot occur is 0. ( Such an event is called impossible event)
 Property 3: The probability of an event that must occur is 1. ( such event called certain event) ex. numbers 5 and 0.23 could not possibly be probabilities.
P(E)  probability of event

Event and Sample space
 Event (E)  a collection of outcomes for the experiment, that is, any subset of the sample space.
 Sample space (S) the collection of all possible outcomes for an experiment
 Specified event occurs  if that event contains the card selected. Ex. if the card selected turns out to be the king of spades, the second and fourth event occur, whereas the first and third events do not.

Van diagrams
 are one of the best ways to portray events and relationships among events visually.
 S  rectangle
 E disks, circles inside the rectangle

Complement of E
 everything ot side of E "not E"
 The event "E does not occur"

A & B
 A intersection B or A B, A and B
 The event " both A and B occur"
 All outcomes common to event A and event B.

A or B
 The event "either A or B or both"
 A union B, A B
 event A or B consists of all outcomes either in event A or in Event B or both. equivalently, that "at least one of event A and B occurs"

Mutually Excllusive Events
 two or more events are mutually exclusive events if not two of them have outcomes in common
 simbol Fee
 intersection is empty
 empty set no elements in it

4.2 Homework problem
 less than 7%  means 7 not included
 At least 8%  8 and more
 write event in parenthesis
 C'  complement of C

Probability Notation
if E is an event, the P(E) represent the probability that event E occurs. Read as " the probability of E"

Special addition rule
 applied only to mutually exclusive events
 P(A or B)= P(A) + P(B)

The complement rule
 P(E)+P(not E)= 1 or
 P(E) = 1P(not E)

General Addition Rule
 used for any event that is not mutually exclusive
 P(A or B) = P(A) + P(B)  P(A&B)
 without using Genral addition rule can be done by f
 N rule
 The general addition rule is consistent with the special addition rule  if two events are mutually exclusive, both rules yeild the same result
 General addition rule for more than 2 events
 ex. for 3 events
 P(A or B or C) = P(A) + P(B) + P(C) P(A&B)  P(A&C)  P(B&C) + P(A&B&C)

P(S)
P(S)= 1 probability of a sample set means all events 36/36=1

