What is the definition of probability?
Probability deals with experiments that yield random short term results or outcomes yet reveal long term predictability.
What is the definition of outcome?
outcomes are random short term results
The long-term proportion in which a certain outcome is observed is________
the probability of that outcome
What does the Law of Large Numbers tell us?
As the number of repetitions of a probability experiment increases, the proportions with which a certain outcome is observed gets closer to the probability of that outcome.
What is the definition of experiment?
Experiment is any process with uncertain results that can be repeated.
What is the definition of sample space?
Sample Space, S of a probability experiment is the collection of all possible outcomes.
What is the definition of an event
An event is any collection of outcomes from a probability experiment. An event consists of one outcome or more than one outcome. We will denote events with one outcome, sometimes called simple events, ei. In general, events are denoted using capital letters such as E.
First two rules of probabilities
1. The probability of any event, E, P(E), must be greater than or equal to zero and less than or equal to one.
2. The sum of the probabilities of all outcomes equals one. That is, if the sample space equals e1, e2, ......en then P(e1)+P(e2)+.......P(En)=1
Probability Model
Probability Model lists all the possible outcomes of a probability experiment and each outcome's probability. A probability model must satisfy rule 1 and 2 of the rules of probabilities.
If an event is impossible, the probability of an event equals_
zero
If an event is a certainty the probability of an event is_
one
An______ is an event that has a low probability of occurring
unusual event
To be an unusual event it must have a probability less than _%
five
The____ and _____ of the problem also determine the probability that separates unusual events from not so unusual events.
researcher, context
Approximating Probabilities Using the Empirical Approach
P(E) ~relative frequency of E~ (frequency of E)/(number of trials of experiment)
Computing Probability using the Classical Method
P(E) = (number of ways that E can occur)/ (number of possible outcomes) or m/n
subjunctive probability
of an outcome is a probability obtained on the basis of personal judgement
two events are _ if they have no outcomes in common.
disjoint
Another name for disjoint events is ____ ______events
mutually exclusive
Example of mutually exclusive events
odds and evens
______ diagrams are used to represent events as circles enclosed in triangles
Venn
Addition Rule for Disjoint Events
If E and F are disjoint events, then
P(E or F) = P(E) + P(F)
The General Addition Rule
For any two events E and F,
P(E or F)= P(E) + P(F)-P(E and F)
Contigency table
also called two-way table because it relates to two categories of data
Compliment of an Event
Let S denote the sample space of a probability experiment and let E denote an event. The compliment of E, denoted E^C is all outcomes in a sample space S that are not outcomes of event E
Compliment Rule
If E represents any event and E^C represents the complement of E, then
P(E^C)=1-P(E)
Two events E and F are ___ if the occurrences of event E in a probability does not affect the probability of event F
independent
two events are ___ if the occurrences of event E in a probability experiment affects the probability of event F
dependent
Disjoint events versus independent events
two events are disjoint if they have nothing in common while two events are independent if one event occurring does not promote the probability of the other event occurring
Multiplication Rule for Independent Events
P(E and F)=P(E)*P(F)
Multiplication Rule for n independent events
P(E1 and E2 and E3) = P(E1)
P(E2)
P(E3)
Conditional Probability Rule
P(F/E)=P(E and F)/P(E0
Conditional Probability
The notation P(F/E) is read the probability of event F given event E. It is the probability that the event F occurs given that the event E has occurred.
General Multiplication Rule
P(E and F) = P(E)*P(F/E)
permutation
a permutation is an ordered arrangement in which r objects are chosen from n distinct objects and repetition is not allowed The symbol nPr represents the number of permutations of r objects selected from n objects.
Knowledge to recall
If small random samples are taken from large populations without replacement, it is reasonable to assume independence of the events. As a rule of thumb, if the sample size is less than five percent the population size, we treat the events as independent.
Two events E and F are independent if
P(E/F)=P(E) and P(F/E)=P(F)
Permutation with non distinct items
The number of permutations of n objects in which n1 are of one kind, n2 are of the second kind, and nk is of the kth kind is given by...
n! divided by n1!
n2!
nk!
combination
is a collection without regard to order in which r objects are chosen from n distinct objects and without repetition. The symbol nCr represents the number of combinations of n distinct objects taken r at a time.