APMA 0650 Terms Test One

Probability

How likely something is to happen

What is the range of probability?

Range of probability is from 0 to 1 (0 - 100%). When adding up probabilities, they must add up to one to be a valid probability model. It cannot be under or over one.

Outcome

One possible result

Event

A single outcome or specific group of outcomes from an experiment

Sample Space

A set of all possible outcomes

What symbol is used to denote a sample space

Curly Brackets - { }

Set

An unordered collection of distinct outcomes

Does it matter the order of outcomes in a set?

No! Ex: {2, 0 , 1} and { 1 , 2, } are the same set bc it contain the same numbers

Notation of probability of an event

P (A) where P = probability and A = event

Intersection

• "probability of A and B occurring” - P(A n B)

  • Both of the events need to happen

Union

• “probability of A or B occurring” - P(A u B)

  • At least one of the events will happen

Complement

• elements that are not in the given set

  • Notation - A^c OR A’

  • P(A^c) - “probability of everything not in A happening)

What is the probability of P(A u A^c)

Probability is one! It adds everything together in the sample space.

What is the probability of P(A n A^c)

Probability is equal to zero! A and A^c do not share anything together

(A u B)’ = …

(A u B)’ = A’ u B’

(A n B)’ ..

(A n B)’ = A’ n B’

Independence

Two events are independent if one event does not impact the other event 

Conditional probability

• “probability of A given that B” - P(A | B)

  • Prob of A given that we already know that event B happened

Bayes Theorem Equation

Equation for Conditional Probability

• P(A | B) = P(A n B)/ P(B)

  • P(B | A ) = P (B n A)/ P(A)

Subset

•  Every outcome in set A is also in B

• Notation = A ⊆ B -> “A is a subset of B”

• Ex: A = {1,2,3}, B = {1,2,3,4,5} -> in this case, A is a subset!

If If A ⊆ B , then P(A) ??? P(B)?

• If A ⊆ B , then P(A) ≤ P(B) for all events A and B in the sample space

  • A has less or equal elements than B

S^n

• S^n is the set of all sequences of length n using the elements in S.

  • S2 for if you flip a coin twice = {HH, HT, TH, TT}

X c A

x is contained in A, meaning that x is one of the outcomes in set A 

Mutually Exclusive

Two or more outcomes that cannot happen at the same time

• P(A or B) = P(A) + P(B)

• Intersection is zero (A n B) = 0

Probability Distribution

Assignment of numerical probabilities to all the events in the sample space