Probability Flashcards
Denotation:
Event
Probability
A –> Event
P(A) -> Probability –> Preferred outcome / all outcomes
P of combined events, for example ace of spades?
P(ace of spades) = P(ace). P(spades)
Experimental probabilities vs. theoretical probabilities
Experimental is out own experience / observation –> Much easier to compute
Theoretical is hard to compute
Calculate experimental probability
P(A) = successful trials / all trials
Expected value after running the experiment n times
E(A) = P(A) * n
P(7) for a dice
P[E(A)] = P(7) = 1/6
Chance of an expected output of sum 7 in two throws = 6/36
Probability frequency distribution
Collection of the probabilities for each possible outcome
A complement of an event is?
Everything the event is NOT
The Complement of event A is noted as
A’
(A’)’ =
A
P(A’) =
1 - P(A)
What do Combinatorics do?
Deals with the combination of objects from a finite set
What are the 3 integral parts of combinatorics:
Permutations
Variations
Combinations
What are Permutations?
The number of different possible ways we can arrange a set of elements
3 podium spots over 3 drivers
What is P(3) ?
And what is the result called?
6 ways
These 6 ways are called permutations
What are Variations?
Total nr of ways we can pick and arrange elements of a given set
Combination lock with 2 rings with possible outcomes A, B, C?
Possibilities + what is the result called?
3 * 3 = 9
Called variations
Explain Permutations, Combinations and Variations
Permutations = The number of different possible ways we can arrange a set of elements
(you can arrange 3 people in 6 ways, 6 permutations)
Combinations = Number of different ways we can pick certain elements of a set
(there is 120 combinations when picking 3 people out of 10 people)
Variations = Total nr of ways we can pick and arrange elements of a given set
(Product of permutations times combinations –> 6 * 120 = 720)
Explain Permutations, Combinations and Variations with formulas
C = V / P
Oftewel: C = n! / p!(n-p)!
Permutations with repetition –>
Permutations without repetition –>
Combinations without repetition –>
Combinations with repetition –>
Permutations with repetition –> n^p
Permutations without repetition –> n! / (n-p)!
Combinations without repetition –> n!/(p!(n-p)!)
Combinations with repetition –> (n+p-1)! / (p!(n-1)!)
Explain Event and Set
Event has a Set of outcomes –> These are the favourable outcomes as mentioned earlier
Notation Set and Elements
Set –> UPPER CASE
Elements –> lower case
How to denote empty set
EMPTY –> null set or empty set –> Ø
What can non-empty sets can be:
Finite, Infinite
The symbol to note whether an element is part of a set:
x ∈ A
Reads as: “x in A”
Where x = element
Where A = SET
Other way round. If we want to state that the set A has en element x
“A ∋ x”
Reads as: “A contains x”
Element is NOT contained in a set?
x ∉ A
Reads as: “x is not in A”
Generalized statement about multiple elements
∀
Reads as:”for all / any”
“∀x∈A”
Meaning?
Reads as “for all x in A”
What does a Colon mean?
Statements about a specific group of elements within a set
∀ x ∈ A : x is even
Reads as: “for all x in A, such that, x is even
Explain Subset
If every element of A is also an element of B, then A is a subset of B
Notation:
A ⊆ B
Explain Union:
A combination of all outcomes preferred for either A or B:
A⋃B
Remember ⋃ stands for Union
Intersection: Objects that belong to set A and set B
Union: Objects that belong to set A or set B
Give union and intersection of A,B in case of:
No intersection
A ⋂ B = Ø
A ⋃ B = A + B (no double counting)
Give union and intersection of A,B in case of:
Intersection
A ⋃ B = A + B - A ⋂ B
Union = Set A + Set B minus the intersection of A and B
Give union and intersection of A,B in case of:
B is a subset of A
A ⋃ B = A
A contains all values that are also in B
A ⋂ B = B
The intersection is equal to the smaller subset
Explain Mutually exclusive sets
Sets which are not allowed to have any overlapping elements
Mutually exclusive sets have the empty set as their intersection
The probability of getting A, is we are given that B has occured denoted as?
P (A | B)
Read as: “P of A given B”
Say A = Queen and B = Spades
What is P(A | B) ?
And what is this probability called?
Conditional Probability. The likelihood of an event occuring assuming a different one has already happened
Independent events notation?
P(A) = P(A | B)
Als A hetzelfde is ondanks event B, dan is A onafhankelijk van B
Conditional Probability formula?
P(A | B) = P(A ⋂ B) / P(B)
(De intersectie gedeeld door kans B is gelijk aan kans A in geval van B)
Voorbeeld:
A = Schoppen Vrouw
B = Schoppen
Oftewel:
1/13 = (1/52) / (1/4)
Is P(A | B) the same as P(B | A)
Absolutely NOT
P(A ⋃ B) = ?
P(A) + P(B) - P( A ⋂ B)
P( A ⋂ B) = ?
P( A ⋂ B) = P(A) + P(B) - P(A ⋃ B)
Kaarten voorbeeld: Wat is de kans dat je bij de 2e kaart schoppen pakt –> P(A)
P(B) –> Geen Schoppen
1e keer: 13/52
2e keer: 13/51
13/52 * 13/51 = 0.191
1e keer: P(B) = 0.75 –> Alles behalve schoppen
2e keer: P(A | B) –> 0.255 –> 13/51 –> Mist 1 kaart
What is Bayes Law?
Follows from?
Bayes Law: Allows us to find a relationship between the different conditional probabilities of two events
P(A | B) = P(B | A) * P(A) / P(B)
Volgt logischerwijs uit:
P(A | B) = P(A ⋂ B) / P(B)
Waarbij P(A ⋂ B) is vervangen door P(B | A) * P(A)
