Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability an event occurs regardless of values of the other event"
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability an event occurs regardless of values of the other event"
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability an event occurs regardless of values of the other event"
Marginal probability
P(A) or P(B)
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability an event occurs regardless of values of the other event"
Marginal probability
P(A) or P(B)
The probability that a randomly selected student in STA199 favors dogs over cats.
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability two or more events simultaneously occur"
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability two or more events simultaneously occur"
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability two or more events simultaneously occur"
Joint probability
P(A∩B)
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability two or more events simultaneously occur"
Joint probability
P(A∩B)
The probability that a randomly selected student in STA199 favors dogs and their favorite movie genre is drama.
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability an event occurs given the other has occurred"
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability an event occurs given the other has occurred"
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability an event occurs given the other has occurred"
Conditional probability
P(A|B)=P(A∩B)P(B) or P(B|A)
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability an event occurs given the other has occurred"
Conditional probability
P(A|B)=P(A∩B)P(B) or P(B|A)
The probability that a randomly selected student in STA199 favors dogs given their favorite movie genre is drama.
Q - Find the correct term, a mathematical notation with events A and B, and an example for each of the following descriptions.
"The probability an event occurs given the other has occurred"
Conditional probability
P(A|B)=P(A∩B)P(B) or P(B|A)
The probability that a randomly selected student in STA199 favors dogs given their favorite movie genre is drama.
A given B restricts our attention to B from the whole sample space Ω.
Q - What is the multiplicative rule of probability?
P(A∩B)=P(A|B)P(B)
Comes directly from definition of conditional probability.
Q - What is the multiplicative rule of probability?
P(A∩B)=P(A|B)P(B)
Comes directly from definition of conditional probability.
Q - What is the law of total probability?
Q - What is the multiplicative rule of probability?
P(A∩B)=P(A|B)P(B)
Comes directly from definition of conditional probability.
Q - What is the law of total probability?
P(A)=P(A|B)P(B)+P(A|Bc)P(Bc)=P(A∩B)+P(A∩Bc)
Let's verify it with a Venn Diagram!
Q - What are the independent events?
Q - What are the independent events?
Knowing one event has occurred does not lead to any change in the probability we assign to another event.
P(A|B)=P(A) or P(B|A)=P(B)
P(A∩B)=P(A)P(B) if A and B are independent.
Q - What are the independent events?
Knowing one event has occurred does not lead to any change in the probability we assign to another event.
P(A|B)=P(A) or P(B|A)=P(B)
P(A∩B)=P(A)P(B) if A and B are independent.
P(animal|movie) = P(animal)
Knowing one's movie taste tells us nothing about his/her preference of dogs over cats.
Q - Are two disjoint events with positive marginal probability independent?
No
P(A)P(B)>0 while P(A∩B)=0 and thus P(A∩B)≠P(A)P(B).
Q - Are dying and abstaining from coffee independent events?
Let A be the event that a man died and B be the event that a man was a non-coffee drinker.
| Did not die | Died | |
|---|---|---|
| Does not drink coffee | 5438 | 1039 |
| Drinks coffee occasionally | 25369 | 4440 |
| Drinks coffee regularly | 24934 | 3601 |
Q - Are dying and abstaining from coffee independent events?
Let A be the event that a man died and B be the event that a man was a non-coffee drinker.
| Did not die | Died | |
|---|---|---|
| Does not drink coffee | 5438 | 1039 |
| Drinks coffee occasionally | 25369 | 4440 |
| Drinks coffee regularly | 24934 | 3601 |
Q - Are dying and abstaining from coffee independent events?
Let A be the event that a man died and B be the event that a man was a non-coffee drinker.
| Did not die | Died | |
|---|---|---|
| Does not drink coffee | 5438 | 1039 |
| Drinks coffee occasionally | 25369 | 4440 |
| Drinks coffee regularly | 24934 | 3601 |
Q - Are dying and abstaining from coffee independent events?
Let A be the event that a man died and B be the event that a man was a non-coffee drinker.
| Did not die | Died | |
|---|---|---|
| Does not drink coffee | 5438 | 1039 |
| Drinks coffee occasionally | 25369 | 4440 |
| Drinks coffee regularly | 24934 | 3601 |
Q - What is the probability that a man was a non-coffee drinker given he died?
Q - What is the probability that a man was a non-coffee drinker given he died?
P(B|A)=P(B∩A)/P(A)=1039/9080=0.114
Q - What is the probability that a man died given he was a non-coffee drinker?
Q - What is the probability that a man died given he was a non-coffee drinker?
P(A|B)=P(A∩B)/P(B)=1039/6477=0.160
Q - P(A|B) and P(B|A) are not the same. Are they related in some way?
Yes - According to Bayes' theorem,
P(A|B)=P(B|A)P(A)P(B). If we know marginal probabilities and one conditional probability, we can always compute the other conditional probability!
This can be viewed as updating prior belief P(A) to posterior belief P(A|B) after seeing new information B.
Q - Let's recompute P(A|B) using Bayes' theorem.
Q - Let's recompute P(A|B) using Bayes' theorem.
P(A|B)=P(B|A)P(A)P(B)=1039/9080×9080/648216477/64821=10396477=0.160
Q - What is Simpson's paradox?
The effect where the inclusion of a third variable in the analysis can change the apparent relationship between the other two variables
Q - What is Simpson's paradox?
The effect where the inclusion of a third variable in the analysis can change the apparent relationship between the other two variables
Q - Explain Simpson's paradox in UC Berkeley admission data in mathematical expressions.
Q - What is Simpson's paradox?
The effect where the inclusion of a third variable in the analysis can change the apparent relationship between the other two variables
Q - Explain Simpson's paradox in UC Berkeley admission data in mathematical expressions.
Overall, P(Admit|Female) < P(Admit|Male)
However, P(Admit|Dept, Female) ≈ P(Admit|Dept, Male) or even P(Admit|Dept, Female) > P(Admit|Dept, Male)
Q - The effect of the function group_by() applies to all the subsequent operations. What function do we need to undo grouping?
Q - The effect of the function group_by() applies to all the subsequent operations. What function do we need to undo grouping?
ungroup()
summarize() also peels off one layer of grouping by default; if there were multiple grouping variables, then the dataset would still be grouped even after summarize.
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