Bayesian Inference Flashcards
“x e A” reads
“x in A”
x is an element of the set A
What is does the symbol of “upside down A” read
For all/ any
What does the “:” semi-colon read?
such that
it is incredibly useful when we want to make statements about a specific group of elements within a set
What is the definition of a subset?
a subset is a set that is fully contained in another set
T or F - If every element of A is also an element of B then A is a subset of B
True
What is this symbol? ⊆
subset symbol
What are the 2 subsets that every set contains?
The set itself
and the null set
Which of the following is true about events?
a) There always exists a set of outcomes that satisfy a given event.
b) The outcomes of an event always have to be numerical.
c) We denote events with lower-case letters of the Latin alphabet like “x” or “y”.
d) We denote individual solutions to these events with upper-case letters from the Latin alphabet like “X” or “Y”.
a) There always exists a set of outcomes that satisfy a given event.
What are the 3 ways events can interact or be represented?
- Never touch
- Intersect
- Completely overlap
What happens when the event circles never touch?
The two events can never happen simultaneously
Event A occurring guarantees that event B is not occurring and vice versa
If two events (circles) intersect, can these events occur at the same time?
Yes
Which of the following does not describe two events with overlapping sets of outcomes?
Being European and being male.
Being white and being French.
Eating meat and being vegan.
Living in Ohio and liking country music.
Eating meat and being vegan.
If two events can occur at the same time, then their respective circles would…
completely overlap.
partially overlap (intersect).
not touch at all.
be tangent to one another.
partially overlap (intersect).
When we want set A and set B to happen at the same time, what are we talking about?
Their intersection
What does the intersection of two events contain?
All the outcomes that are favorable for both event A and event B simultaneously
How is two intersecting sets denoted?
A ∩ B
What is the intersection called of two non overlapping sets?
The empty set
A ∩ B = ∅ or {}
There are no outcomes which satisfy both events simultaneously
When do we use intersections?
only to denote when instances where both events A and B happen simultaneously
Mark the FALSE statement. If we know that “a is in A” and “b is in the intersection of A and B”, then:
A contains a
b is part of the intersection of B and A
B contains b
a is part of B
a is part of B
Mark the pair of sets, whose intersection is NOT the null set.
Odd numbers and even numbers.
Non-positive numbers and non-negative numbers.
Negative and positive numbers.
Numbers divisible by 2 and odd numbers.
Non-positive numbers and non-negative numbers.
Which of the following showcases the two sets, whose intersection is the smaller set.
Black cards and Clubs.
Aces and Jacks.
Queens and Spades.
Diamonds and Hearts.
Black cards and Clubs.
Set A and set B intersect. What happens if we require only one set to occur? What is this called?
A union
What is the union of a set?
a combination of all outcomes preferred for either A or B
Example, if you think of the US as the Union, each state as the Set, then each Element would be what?
The citizens of each state
How is the union of two sets denoted?
A∪B
What is the formula for unions?
The union of A and B = A + B - their intersection
A ∪ B = A + B - A ∩ B
If the intersection of two sets, A and B, is the smaller set, then their union is….. :
The larger set
The same as their intersection
The sum of the two sets
None of the above
The larger set
There are 8 blond and 10 brown-eyed people in the office. If only Jason and Eve have both features, how many people represent the union of blond and brown-eyed people?
18
16
14
20
16
What are mutually exclusive sets?
Sets which are not allowed to have any overlapping elements
*graphically their circles never intersect
What is the intersection of mutually exclusive sets?
The empty set
also, if the empty set is the intersection of two sets then they must be mutually exclusive
What is the union of two mutually exclusive sets?
the sum of their sets
A ∪ B = A + B
Do sets have complements?
Yes
What are complement sets?
All values that are part of the sample space but not part of the set
T or F
Complements are always mutually exclusive
True
Are all mutually exclusive sets complements?
No
How do we graphically express two mutually exclusive sets?
Their circles are tangent to one another.
Their circles do not touch.
Their circles intersect.
The circle of one completely overlaps the circle of the other one.
Their circles do not touch.
What are the intersection and union of two mutually exclusive sets?
The intersection is their sum and the union is the empty set.
The intersection is the smaller set and the union is the larger set.
The union is the empty set and their sum is the intersection.
The union is their sum and the intersection is the empty set.
The union is their sum and the intersection is the empty set.
Which is the correct statement?
All mutually exclusive sets are complements.
No mutually exclusive sets are complements.
All complements are mutually exclusive.
No complements are mutually exclusive.
All complements are mutually exclusive.
Which pair of sets are mutually exclusive, but not complements?
Even numbers and prime numbers.
Winning and Drawing (ending in a draw).
Red cards and Black cards.
Heads and Tails.
Winning and Drawing (ending in a draw).
What are Independent Events?
events where the theoretical probability remains unaffected by other events
What are Dependent events?
events where their probabilities vary as conditions change
How is this notation read? P(A|B)
P of A given B
What is used to distinguish between dependent and independent events?
The Conditional Probability
If the probability of an event remains unaffected by another event, the two are….
Dependent.
Independent.
Mutually Exclusive.
None of the above.
Independent.
Going back to the card example, which of the following sets of events are independent?
Drawing a Queen and drawing a Jack.
Drawing a Heart and drawing the Jack of Hearts.
Drawing a Diamond and drawing an Ace.
Drawing a four and drawing the Ace of Spades.
Drawing a Diamond and drawing an Ace.
What do we call the probability we use to distinguish dependent from independent events?
The dependent probability.
The independent probability.
The conditional probability.
The distinguishing probability.
The conditional probability.
What is the definition of Conditional Probabilities?
The likelihood of an event occurring, assuming a different one has already happened
How do you compute and interpret Conditional Probability?
P(A|B) = P(A ∩ B) / P(B)
What is the “Favorable over all” formula?
P(A) = favorable
————
all
-
Does the order in which we write the elements in conditional probability matter?
Yes - it is crucial
P(A|B) /= P(B|A)
What can you conclude about events A and B given that P(A) = P(A|B)?
The two are independent.
The two are dependent.
Event “A” is more likely to occur.
Event “B” is more likely to occur.
The two are independent.
Applying the Conditional Probability Formula, what is the probability of event A occurring, given event B has occurred if the likelihood of getting their intersection is 0.15 and the likelihood of event B is 0.6?
0.45
0.25
4
0.75
0.25
What is the difference between P(A|B) and P(B|A)?
The former suggests the two events are dependent, while the latter suggests they are independent.
The former suggests event A is more likely than event B, while the latter suggests B is the likelier of the two.
One indicates the probability of getting A, given B has occurred, while the other indicates the likelihood of getting B, given A has occurred.
All of the above.
One indicates the probability of getting A, given B has occurred, while the other indicates the likelihood of getting B, given A has occurred.
What is the Law of Total Probability?
It is a formula that can express the total probability an outcome which can be realized by several distinct events
https://en.wikipedia.org/wiki/Law_of_total_probability#Example
What is the formula for the Additive Law
P(A∪B) = P(A) + P(B) - P(A∩B)
The probability of the union of two sets is equal to the sum of the individual probabilities of each event minus the probability of their intersection
Can we rearrange the Additive Law Formula?
To get the intersection we can change to:
P(A∩B) = P(A) + P(B) - P(A∪B)
Calculate P(B U A), given that P(A) = 0.75, P(B) = 0.6 and P(B Intersect A) = 0.55.
0.8
0.7
0.4
0.(81)
0.8
Use the Additive Law to find the probability of the intersection of A and B, given that P(B U A) = 0.9, P(A) = 0.65, P(B) = 0.44
0.19
0.69
0.99
0.41
0.19
What is the Multiplication Rule?
P(A|B) x P(B) = P(A∩B)
How do we interpret the multiplication rule for events A and B?
The probability of both events occurring simultaneously equals the product of the likelihood of A occurring and the conditional probability that B occurs, given A has already occurred.
The probability of both events occurring simultaneously equals the product of the likelihood of A occurring and the conditional probability that A occurs, given B has already occurred.
The probability of both events occurring simultaneously equals the product of the likelihood of B occurring and the conditional probability that B occurs, given A has already occurred.
None of the above.
The probability of both events occurring simultaneously equals the product of the likelihood of A occurring and the conditional probability that B occurs, given A has already occurred.
Imagine you are watching your favorite British soccer team and want to know how likely it is that they will win, and your favorite player Mohamed Salah will score. You know that this season, the team has won in 80% of the games he has scored in. Furthermore, you know that he has scored in 70% of all the games the team has played this season.
0.50
0.56
0.10
0.73
0.56
What is Bayes’ Rule? or Bayes’ Theorem?
P(A|B) = P(B|A) x P(A) / P(B)
It allows us to find a relationship between the different conditional probabilities of two events
What is one of the most prominent places of use for Bayes’ Theorem?
Medical Research
- Trying to find a causal relationship between symptoms
- Helps us make more reasonable arguments about which one causes the other
We know that out of those 45% that had relevant experience, 50% also performed well academically. How would you denote this?
P( Grades | Experience ) = 50%
According to the lecture, what is one field where Bayes’ Theorem is often used?
Medicine
Logistics
Agriculture
Sports
Medicine
What is the value of P(A|B), knowing P(B|A) = 0.6, P(A) = 0.4 and P(B) = 0.3?
0.24
0.2
0.8
0.(6)
0.8
