Fundamentals of Probability Flashcards
What is a sample space
It is a collection of all possible outcomes of an experiment, denoted as “S”
What is an event
it is any collection of outcomes of the experiment denoted as “A”, “B”, etc.
Includes individual outcomes, the null set, the entire sample space
What do we call an outcome that is a member of an event
The is event is said to have occurred
What is probability
A scientific way to measure uncertainty
What is a set?
is a collection of objects. Denoted with capital letter. for example A={a,b,c,…,z}
What is an element
An element is a part of a set. for example z epsilon A says z is an element of a set A
What is a countable set
is a set where you can count the number of elements in a set. In other words if you can assign a natural number to the elements of the set then it is countable
What is an empty set/null set
A set that doesn’t have elements { } or 0crossed out. An empty set is always a subset of every set.
What are the types of countable sets
Finitely Countable or Infinitely countable.
Finitely countable set
D= {0,1,2,…10]
Infinitely countable set
The set of natural numbers N = {1,2,3,…}
What’s the subset concept
A is a subset of B if every element in A is also in B. for example A={x,y,z} B={w,x,y,z} A (c underlined) subsest B
How do you calculate the number of subsets in a set
2^n where n is the number of elements
Equal sets
A and B are equal if A c underline (subset) B and B c underline (subset) of A
Intersection of event
it is the “and” of two events. for example AB is a set that contains elements in both A and B. for example
A={2,4,6} B = {4,5,6,7,8}
AB = {4,6}
Union of events
it is the “or”.
AUB is a set that contains elements in either A or B. For example AuB={2,4,6,7,8} when we have A={2,4,6} B = {4,5,6,7,8}
A complement or exhaustive
is an element outside of the set. We usually use exhaustive
If AcB then…
AUB=B
A={1,2,3} B={1,2,3,4,5,6}
AUB={1,2,3,4,5,6}
If AcB and BcA then…
then A=B
A={1,2,3} B={1,2,3}
If AcB then…
then AB=A
A={1,2,3} B={1,2,3,4,5,6} then AB={1,2,3}
if AUAc then
= S (the sample space). A union of A complement
A and B are mutually exclusive it..
they don’t have any outcomes in common. aka disjoint
Their union is equal to the sample space, but they have no events in common.
Groups of events that are mutually exclusive and exhaustive
A and B are mutually exclusive and exhaustive
AUB = S but they have no events in common. It means that they make up the entire sample space but have no common events
A probability on a sample S is
We assign every event A a number P(A), which is the probability the event will occur (P:S—>R).
A collection of numbers P(A) that satisfy these axioms:
1. P(A) >=0 for all AcS
2. P(S) =1
3. For any sequence of disjoint sets (A1, A2, A3,…, this could be an infinite sequence) The probability of the union of that sequence is equal to the probabilities of those events. P(UiAi)=SigmaiP(Ai)
P(Ac) =
the probability of A compliment
1-P(A)
P( ) =
probability of empty set
0
If AcB then
P(A)<=P(B)
For all A
0<=P(A)<=1
P(AUB)=
P(A) + P(B) - P(AB)
P(ABc) =
probability of AB compliment
P(A)-P(AB)
What is the definition of Combinatronics
Counting Rules - fancy counting
What are the combinatronics
- If an experiment has two parts, first one having m possibilities and regardless of the outcome in the first part, the second one having n possibilites, then the experiment has m*n possible outcomes
- An ordered arrangement of objects is called a permutation. The number of different permutations of N objects is N!. The number of different permutations of n objects taken from N objects is N!/(N-n)!
- Any unordered arrangement of objects is called a combination. The number of different combinations of n objects taken from N objects is N!/{N-n)!n!}. this is typically denoted as (Nn)—>”N choose n”
Permutation
any ordered arrangement of objects. The number of different permutations of N objects is N!. The number of different permutations of n objects taken from N objects is N!/(N-n)!
Combination
Any unordered arrangement of objects. The number of different combinations of n objects taken from N objects is N!/{N-n)!n!}. we call this N choose n and it is denoted as
(N
n)
A \ B is
This reads A set minus B and means a set containing all elements of A which are not elements of B.
Independence
Events A and B are independent if the probability of their intersection is equal to the product of their probability in an equation it read P(AB) = P(A)*P(B)
If A and B are independent, then A and B complete are
also independent
Probability of A conditional on B is P(A|B) and is
P(AB)/P(B) assuming P(B)>0
T
Baye’s Theorem
P(A|B) = P(B|A)P(A)/{P(B|A)(P(A) + P(B|A complement)P(A complement). As long as A and A complement form a partition of S
Prior probability is also known as
the unconditional probability
