Week 1 Notes Flashcards
Probability theory
Is a mathematical model for chance or random phenomena
Sample space
The sample space for a probability/random experiment (I.e. an experiment with random outcomes) is the set of all possible outcomes
The sample space is denoted S
An outcome is an element of S, generally denoted s is an element of S
Event
An event is a particular subset of the sample space. Events are typically denoted with capital letters- A, B, C, etc. - and we write A is a subset of S to indicate that A is an event of S
When is a probability measure valid?
If it is a function P from subsets of the sample space S to the real numbers such that the following hold:
1: P(1)=1
2: if A is any event in S, then P(A) >= 0
3: if events A1 and A2 are disjoint, then
P(A1 U A2) = P(A1) + P(A2)
Let S be a sample space with probability measure P. Also, let A and B be any events in S. What five properties hold?
1: P(A complement) = 1 - P(A)
2: P(empty set) = 1
3: If A is a subset of B, then P(A)<=P(B)
4: P(A) <= 1
5: P(A U B) = P(A) + P(B) - P(A intersect B)
Relative frequency approximation
P(A) is approximately
times A occurred) / (# trials
What is the law of large numbers
As the number of trials increases, the relative frequency approximation approaches the theoretical value of P(A)
What is the multiplication principle?
If one experiment has n1 outcomes and another experiment has n2 outcomes, then there are n1 x n2 total outcomes for the composite of the two experiments. More generally, if there are m experiments with the first experiment having n1 outcomes, the second with n2, etc., then there n1 x n2 x … x nm total outcomes for the composite of the m experiments.
What is a permutation?
A permutation is an ordered arrangement of objects. For example, “MATH” is a permutation of four letters from the alphabet.
What is a combination?
A combination is an unordered collection of r objects from n total objects. For example, a group of 3 students chosen from a class of 10 students
How can permutations be counted?
The number of permutations of r objects from n distinct objects without replacement is given by:
nPr = (n!)/((n-r)!) = n x (n-1) x … x (n-r+1)
How can combinations be counted?
The number of combinations of r objects from n distinct objects without replacement is given by:
nCr = (r ; n) = (n!)/((r!)(n-r)!)
Note that n chose r is also referred to as a binomial coefficient, since it appears in the binomial expansion theorem
