Probability

Question 1

Denotation:
Event
Probability

Answer 1

A –> Event

P(A) -> Probability –> Preferred outcome / all outcomes

Question 1

P of combined events, for example ace of spades?

Answer 1

P(ace of spades) = P(ace). P(spades)

Question 2

Experimental probabilities vs. theoretical probabilities

Answer 2

Experimental is out own experience / observation –> Much easier to compute

Theoretical is hard to compute

Question 3

Calculate experimental probability

Answer 3

P(A) = successful trials / all trials

Question 4

Expected value after running the experiment n times

Answer 4

E(A) = P(A) * n

Question 5

P(7) for a dice

Answer 5

P[E(A)] = P(7) = 1/6
Chance of an expected output of sum 7 in two throws = 6/36

Question 6

Probability frequency distribution

Answer 6

Collection of the probabilities for each possible outcome

Question 7

A complement of an event is?

Answer 7

Everything the event is NOT

Question 8

The Complement of event A is noted as

Answer 8

A’

Question 9

(A’)’ =

Answer 9

A

Question 10

P(A’) =

Answer 10

1 - P(A)

Question 11

What do Combinatorics do?

Answer 11

Deals with the combination of objects from a finite set

Question 12

What are the 3 integral parts of combinatorics:

Answer 12

Permutations
Variations
Combinations

Question 13

What are Permutations?

Answer 13

The number of different possible ways we can arrange a set of elements

Question 14

3 podium spots over 3 drivers

What is P(3) ?
And what is the result called?

Answer 14

6 ways
These 6 ways are called permutations

Question 15

What are Variations?

Answer 15

Total nr of ways we can pick and arrange elements of a given set

Question 16

Combination lock with 2 rings with possible outcomes A, B, C?

Possibilities + what is the result called?

Answer 16

3 * 3 = 9

Called variations

Question 17

Explain Permutations, Combinations and Variations

Answer 17

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)

Question 18

Explain Permutations, Combinations and Variations with formulas

Answer 18

C = V / P

Oftewel: C = n! / p!(n-p)!

Question 19

Permutations with repetition –>
Permutations without repetition –>
Combinations without repetition –>
Combinations with repetition –>

Answer 19

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)!)

Question 20

Explain Event and Set

Answer 20

Event has a Set of outcomes –> These are the favourable outcomes as mentioned earlier

Question 21

Notation Set and Elements

Answer 21

Set –> UPPER CASE
Elements –> lower case

Question 22

How to denote empty set

Answer 22

EMPTY –> null set or empty set –> Ø

Question 23

What can non-empty sets can be:

Answer 23

Finite, Infinite

Question 24

The symbol to note whether an element is part of a set:

Answer 24

x ∈ A

Reads as: “x in A”

Where x = element

Where A = SET

Question 25

Other way round. If we want to state that the set A has en element x

Answer 25

“A ∋ x”

Reads as: “A contains x”

Question 26

Element is NOT contained in a set?

Answer 26

x ∉ A

Reads as: “x is not in A”

Question 27

Generalized statement about multiple elements

Answer 27

∀

Reads as:”for all / any”

Question 28

“∀x∈A”

Meaning?

Answer 28

Reads as “for all x in A”

Question 29

What does a Colon mean?

Answer 29

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

Question 30

Explain Subset

Answer 30

If every element of A is also an element of B, then A is a subset of B

Notation:

A ⊆ B

Question 31

Explain Union:

Answer 31

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

Question 32

Give union and intersection of A,B in case of:

No intersection

Answer 32

A ⋂ B = Ø
A ⋃ B = A + B (no double counting)

Question 33

Give union and intersection of A,B in case of:

Intersection

Answer 33

A ⋃ B = A + B - A ⋂ B

Union = Set A + Set B minus the intersection of A and B

Question 34

Give union and intersection of A,B in case of:

B is a subset of A

Answer 34

A ⋃ B = A

A contains all values that are also in B

A ⋂ B = B

The intersection is equal to the smaller subset

Question 35

Explain Mutually exclusive sets

Answer 35

Sets which are not allowed to have any overlapping elements

Mutually exclusive sets have the empty set as their intersection

Question 36

The probability of getting A, is we are given that B has occured denoted as?

Answer 36

P (A | B)
Read as: “P of A given B”

Question 37

Say A = Queen and B = Spades

What is P(A | B) ?
And what is this probability called?

Answer 37

Conditional Probability. The likelihood of an event occuring assuming a different one has already happened

Question 38

Independent events notation?

Answer 38

P(A) = P(A | B)

Als A hetzelfde is ondanks event B, dan is A onafhankelijk van B

Question 39

Conditional Probability formula?

Answer 39

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)

Question 40

Is P(A | B) the same as P(B | A)

Answer 40

Absolutely NOT

Question 41

P(A ⋃ B) = ?

Answer 41

P(A) + P(B) - P( A ⋂ B)

Question 42

P( A ⋂ B) = ?

Answer 42

P( A ⋂ B) = P(A) + P(B) - P(A ⋃ B)

Question 43

Kaarten voorbeeld: Wat is de kans dat je bij de 2e kaart schoppen pakt –> P(A)

P(B) –> Geen Schoppen

Answer 43

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

Question 44

What is Bayes Law?
Follows from?

Answer 44

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)