Quiz 1 Flashcards
What is a sample space?
set whose elements describe possible outcomes in an experiment
What is an event?
an event A occurs if the outcome of an experiment is an element of the set A
When are two events disjoint?
disjoint if A∩B=∅.
They do not have any outcome in common.
We may equivalently say that they are mutually exclusive.
When does an event A imply B?
an event A implies an event B if A ⊂ B
All the elements of A belong to B
What is De Morgan’s Law?
(A∪B)^C=A^C ∩ B^C
(A∩B)^C=A^C ∪ B^C
What is probability?
A probability P is an assignment of a real value to each event A in a sample space Ω, such that:
- P(A) ∈ [0,1]
- P(Ω) = 1
- If A and B are disjoint,P(A∪B) =P(A) +P(B)
What is P(A^C)
P(A^C) = 1−P(A)
What is P(A∪B)?
P(A∪B) =P(A) +P(B)−P(A∩B)
What is the probability of an event A when the probability of each outcome is the same?
P(A) =|A| / |Ω|
Counting without replacement
N! / (N−P)!
Counting with replacement
N^P
Selecting an unordered sample from the set
(N choose P) =N! / (N−P)!P!
Conditional Probability of A given B?
P(A|B) =P(A∩B) / P(B)
P(A∩B) =P(A|B)P(B)
If two events A and B are mutually exclusive,A∩B=∅ so P(A∩B) = 0, and according to the equality above,P(A|B) = 0 as one would expect.
Bayes Formula
P(A|B) =P(B|A)P(A) / P(B)
Law of Total Probability
P(A) =N∑i=1 P(A|Bi)P(Bi)