Probability Theory And Stats 114
Permutations
example : how many different arrangements of A,B,C
3!
example: 6 men, 4 women
(a) How many permutations?
10!
(b) How many permutations men together and women together
6! ∗ 2! ∗ 4! = 34560
Overcounting
n factorial divided by groups of objects that repeat factorial
, example: different outcomes of DUDE
4!/2!
Combinations
how many groups of size r can be chosen from a collection of n objects
Combination Formula
The combination formula is used to calculate the number of ways to choose a subset of
items from a larger set, *where the order of selection does not matter.
n!
C(n, r) =
r!(n − r)!
Probabilities
Sample space (S) : Set of all possible outcomes of an experiment
Event space (E) : a subset of the Sample space E c S
Complement : (E ) All outcomes not in an event in the sample space
c
Define Probability
Assume that for any event E in a sample space,there exists a value P(E),
such that the following is satisfied
Axiom 1 : 0 <= P(E)<= 1
Axiom 2: P(S) = 1
for any sequence of mutually exclusive events,E1,E2,E3
P(E1 or E2 or E3 ) = P(E1) + P(E2) + P(E3)
Order of Operations
Compliment
Intersection
Union
Conditional Probability
#definition the conditional probability of an event E, given an event F is:
P (EF )
P (E|F ) =
P (F )
P (E)P (F |E)
P (E|F ) =
P (F )
Permutations
example : how many different arrangements of A,B,C
3!
example: 6 men, 4 women
(a) How many permutations?
10!
(b) How many permutations men together and women together
6! ∗ 2! ∗ 4! = 34560
Overcounting
n factorial divided by groups of objects that repeat factorial
, example: different outcomes of DUDE
4!/2!
Combinations
how many groups of size r can be chosen from a collection of n objects
Combination Formula
The combination formula is used to calculate the number of ways to choose a subset of
items from a larger set, *where the order of selection does not matter.
n!
C(n, r) =
r!(n − r)!
Probabilities
Sample space (S) : Set of all possible outcomes of an experiment
Event space (E) : a subset of the Sample space E c S
Complement : (E ) All outcomes not in an event in the sample space
c
Define Probability
Assume that for any event E in a sample space,there exists a value P(E),
such that the following is satisfied
Axiom 1 : 0 <= P(E)<= 1
Axiom 2: P(S) = 1
for any sequence of mutually exclusive events,E1,E2,E3
P(E1 or E2 or E3 ) = P(E1) + P(E2) + P(E3)
Order of Operations
Compliment
Intersection
Union
Conditional Probability
#definition the conditional probability of an event E, given an event F is:
P (EF )
P (E|F ) =
P (F )
P (E)P (F |E)
P (E|F ) =
P (F )
