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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
- 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.
- 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.
- Equal-likelihood 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.
- 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
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) = 1-P(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)= 1 probability of a sample set means all events 36/36=1