chapter 1 Flashcards
the process which an observation (or measurement) data is obtained through either uncontrolled events in nature or controlled situations in a laboratory.
Experiment
the set whose elements are all the possible outcomes of an experiment.
Sample Space
elements in a sample space.
Sample Points
has a finite number of outcomes.
ex. Outcomes of a single coin tossed
Finite Sample Space
has an infinite number of outcomes.
ex. Waiting time at the bus stop
Infinite Sample Space
a subset of the sample space.
Event
an event that contains one
sample point.
Simple Event
has no outcomes,
cannot occur.
Null Space {} or Empty Set Ø
What are the three (3) set operations?
Union, intersection, complement
Set Operations
The ________ of two events consists of all outcomes that are contained in one event or the other, denoted as 𝑬𝟏∪𝑬𝟐.
Union
Set Operations
The ________ of two events consists of all outcomes that are contained in one event and 𝑬𝟏∩𝑬𝟐.
Intersection
Set Operations
The _____________ of an event is the set of outcomes in the sample space that are not contained in the event, denoted as 𝑬’
Complement
Events A and B are ______________________ or disjoint because they share no common outcomes.
mutually exclusive
The occurrence of one event precludes the occurrence of the other.
mutually exclusive
Symbolically, A ∩B = Ø
mutually exclusive
mutually exclusive
A∪B= B∪A and A∩B= B∩A
Commutative Law (Event Order Is Unimportant)
mutually exclusive
(A ∪B) ∩ C= (A ∩ C) ∪(B∩C)
(A ∩B) ∪ C= (A∪ C) ∩(B∪C)
Distributive Law
mutually exclusive
(A ∪B) ∪ C = A ∪(B ∪ C)
(A ∩B) ∩ C= A ∩(B∩C)
Associative Law (Like In Algebra)
mutually exclusive
(A ∪B)’ = A’∩ B’ the complement of the union is the intersection of the complements.
(A ∩B)’= A’∪B’ the complement of the intersection is the union of the complements
Demorgan’s Law
mutually exclusive
(A’)’ = A
Complement Law
The negation of a disjunction is the conjunction of the negations;
The negation of a conjunction is the disjunction of the negations;
True or false?
throat.
Basic Counting Techniques
There are three special rules, or counting techniques, used to determine the number of outcomes in events. They are :
- Multiplication rule
- Permutation rule
- Combination rule
Basic Counting Techniques
“If a certain experiment can be performed in n1 ways and corresponding to each of these ways another experiment can be performed in n2 ways, then the combined experiment can be performed in n1* n2ways.”
Multiplication rule
Basic Counting Techniques
Arrangement of objects w/ regards to order.
Permutation
Give 4 kinds of permutation
- Linear Permutation (Factorial Method)
- Permutation nPr, Permutation of Subsets.
- Permutation of Similar Objects, Permutation with repetition.
- Circular Permutation
PERMUTATION OF DISTINCT OBJECTS
The number of permutations of n distinct objects is n!
Factorial method
PERMUTATION OF DISTINCT OBJECTS
The number of permutations of n objects taken r at a time is
Permutation (nPr)/ Permutation of Subsets
PERMUTATION OF DISTINCT OBJECTS
The number of permutations of n distinct objects arranged in a circle is ________________.
Circular Permutation
PERMUTATION OF DISTINCT OBJECTS
The number of distinct permutations of n distinct objects of which n1 are of the first kind, n2 of the second kind, …, nk of the kth kind.
Also applicable for partitioning or groupings of all the n objects.
Permutation with Repetition
The _____________ of n objects taken at r at a time, where order does not count, is C(n,r) or nCr.
combination
2 special cases for permutation.
- Clustering/grouping.
- Complement of 1.
Additive rule for permutation/combination
(using phrases “greater/more than” & “less than”)
Exclusive Range
Additive rule for permutation/combination
(using phrases “at most” & “atleast”)
Inclusive Range
_______________ is the likelihood or chance that a particular outcome or event from a random experiment will occur.
Probability
Probability is a number in the ____ interval
[0,1]
A probability of:
1 means __________
0 means _______________
certainty, impossibility
It is a number that is assigned to each member of a collection of events from a random experiment that satisfies the axioms of probability.
Probability
____________ are generated by applying basic set operations to individual events,
Joint events
Probabilities of joint events can often be determined from the probabilities of the individual events that comprise them.
true or false?
True
The ____________________ of an event A given the occurrence of an event B is defined as P(A| B)
Conditional Probability
P(A∩B) = P(B|A)·P(A) = P(A|B)·P(B)
Multiplication Rule
P(F) = P(F | H)∗P(H) + P(F | M) ∗ P(M) + P(F | L) ∗ P(L)
Total Probability Rule
Two events are _______________ if any one of the following equivalent statements is true:
- P(A|B)=P(A)
- P(B|A)=P(B)
- P(A∩B)=P(A)·P(B)
This means that occurrence of one event has no impact on the probability of occurrence of the other event.
independent
________________ (1702-1761) was an English mathematician and Presbyterian minister.
His idea was that we observe conditional probabilities through prior information.
Thomas Bayes
