Class 5 Spring π· Flashcards
What is the definition of probability?
A branch of pure mathematics about numerical description of the likelihood of an event
What is the definition of statistics?
The collection, organization, displaying, analysis, interpretation, and presentation of data
What does probability help quantify?
Uncertainty
What is the difference between probability and statistics?
Probability is about chance; statistics is about data
What is a simple (marginal) probability?
The probability of an event indicating how likely it is to occur
What is the range of probability values?
0.0 to 1.0 (inclusively)
What does a probability of 0.0 indicate?
Impossibility
What does a probability of 1.0 indicate?
Certainty
Define an event in probability.
A single or a set of outcomes from an experiment or process
What is the probability of getting any 1 number with a fair die?
1/6
What does the Law of Large Numbers (LLN) state?
As more observations are collected, the proportion of occurrences converges to the mathematical probability
What is the common misunderstanding of the Law of Large Numbers?
That random processes compensate for past events (gamblerβs fallacy)
What are disjoint (mutually exclusive) outcomes?
Outcomes that cannot happen at the same time
Give an example of a disjoint outcome.
A single coin toss cannot be both heads and tails
What are non-disjoint outcomes?
Outcomes that can happen at the same time
What is the addition rule for disjoint outcomes?
P(A or B) = P(A) + P(B)
What is the general addition rule for non-disjoint outcomes?
P(A or B) = P(A) + P(B) - P(A and B)
Fill in the blank: The probability of an outcome is the proportion of times the outcome would occur if we observed the random process an _______ number of times.
infinite
True or False: The probability of rolling a 1 or a 2 on a fair die is 1/3.
True
What is the probability of selecting a diamond from a standard deck of cards?
13/52 = 0.250
What is the probability of selecting a face card from a standard deck of cards?
12/52 = 0.231
What is the probability of selecting a card that is both a diamond and a face card?
Calculated using the general addition rule, accounting for double counting
What is a common application of probability in statistics?
Planning a sampling strategy to draw research samples from a larger population
What is the probability of getting heads on the 11th toss after getting heads on the first 10 tosses?
0.5
What is the effect of random sampling errors in statistics?
Separates the effect of random errors from the targeted effect of the treatment
What does the notation for set theory specify?
Compound events
What is the probability of it being a Monday?
P(Monday) = 52 / 365
What is the probability of it being September?
P(September) = 30 / 365
What is the probability of it being a Monday and in September?
P(Monday and September) = 4 / 365
What is the probability of it being a Monday or September?
P(Monday or Sept) = 78 / 365 = 0.214 or 21.4%
True or False: The sum of probabilities of two complementary events always adds up to 1.
True
Define complementary events.
Two mutually exclusive events whose probabilities add up to 1.
What is a sample space?
The collection of all possible outcomes of a trial.
What are some examples of sample space?
- Winning or not winning a lottery ticket
- Rolling a die {1, 2, 3, 4, 5, 6}
- Being taller or shorter than someone {T, S}
- Follower count growth scenarios {G, S, RS}
Do the sum of probabilities of two disjoint events always add up to 1?
Not necessarily, there may be more than 2 events in the sample space.
What is the definition of independent events?
Two processes are independent if knowing the outcome of one provides no useful information about the outcome of the other.
Provide an example of independent events.
Flipping a coin and rolling a die.
What is joint probability?
The probability of two events occurring simultaneously.
What is conditional probability?
The probability of one event occurring in the presence of a second event.
What does marginal probability refer to?
The probability of an event irrespective of the outcome of another variable.
How can we find the joint probability of two dependent events?
Using conditional probability.
What is the formula for calculating conditional probability?
P(A|B) = P(A and B) / P(B)
What does the notation P(A|B) represent?
The probability of event A occurring given that event B has already occurred.
Calculate the conditional probability of rolling a die and getting a value less than 4, given that we know the value is an odd number.
P(value<4 | odd number) = 2/3 or ~0.667
If two events are independent, what is the relationship between their probabilities?
P(A and B) = P(A) * P(B)
If the conditional probability of two events is equal to the first event, what does this indicate?
The events are independent.
What is the basic multiplication rule for independent events?
P(A and B) = P(A) * P(B)
What is the basic multiplication rule for dependent events?
P(A and B) = P(A|B) * P(B)
What is the probability that a patient relapsed in the given study?
P(relapsed) = 48 / 72 ~ 0.67
What is the probability that a patient received the antidepressant and relapsed?
P(relapsed and desipramine) = 10 / 72 ~ 0.14
Calculate P(relapse | desipramine).
P(relapse | desipramine) = 10 / 24 ~ 0.42
What is the probability of receiving the antidepressant given that a patient relapsed?
P(desipramine | relapse) = 10 / 48 ~ 0.21
What does it mean if the conditional probability of two events equals the first event?
They are independent events
The inverse is also true.
If two events are independent, what should knowing the outcome of one provide about the other?
No information
What is the formula for the probability that the first die, X, is 1?
P(X = 1) = Favorable / All outcomes = 1/6
What is the probability that the second die, Y, is 1 or 2?
P(Y = 1 or 2) = Favorable / All outcomes = 2/6
What is the formula for conditional probability?
P(Y=1 or 2 | X=1) = P(Y=1 or 2 and X=1) / P(X=1)
How does knowledge about event X affect the probability of event Y if they are independent?
It does not change it
What happens to P(A and B) in the case of disjoint events?
It goes to 0
What does P(A|B) simplify to in the case of independent events?
P(A)
What is the probability that a randomly selected student is a social science major given that they are female?
P(SS | F) = 0.6
What is the definition of probability?
The proportion of times an outcome would occur if observed infinitely
What values can probability take?
Between 0.0 and 1.0 inclusively
What percentage range can probability also be displayed as?
Between 0% and 100%
What are marginal probabilities?
The probability of a single variable occurring
What are joint probabilities?
Involve outcomes for two or more variables
What is the addition rule for joint outcomes?
P(A or B) = P(A) + P(B) - P(A and B)
What are the conditions for using the simple multiplication rule for joint probability with independent variables?
Events must happen at the same time and be independent
What is the general multiplication rule for dependent variables?
P(A and B) = P(A|B) * P(B)
What is the effect of random errors on variation in populations?
Effect of observable factors + Effect of unavoidable random errors
True or False: Statistics is about data and the βmessy realβ world.
True
Fill in the blank: Probability is about _______ and the βpure mathematicsβ world.
Chance
What does P(SS | M) equal in the context of gender and major dependence?
0.6
What is the implication if P(SS | F) = P(SS)?
Major does not depend on gender
What percentage of all respondents said widespread gun ownership protects citizens?
58%
What was the percentage of White respondents who said gun ownership protects citizens?
67%
What was the percentage of Black respondents who said gun ownership protects citizens?
28%
What was the percentage of Hispanic respondents who said gun ownership protects citizens?
64%
What type of relationship is indicated if opinion on gun ownership and race ethnicity are not independent?
Dependent
