Probability
What is the definition of probability?
The likelihood of an event occurring, often expressed as a fraction or percentage.
Define a sample space in probability.
The set of all possible outcomes of a random experiment.
How many ways can you arrange the letters in the word "STATISTICS"?
11! (factorial)
What is the formula for conditional probability?
P(A | B) = P(A and B) / P(B)
What is the difference between a discrete and a continuous random variable?
Discrete random variables take on a countable number of distinct values, while continuous random variables can take on any value within a range.
If two events A and B are independent, how is the probability of both events occurring calculated?
P(A and B) = P(A) * P(B)
What is a random variable?
A variable that can take on different values based on the outcome of a random experiment.
What is the formula for calculating combinations?
C(n, r) = n! / [r! * (n - r)!]
If events A and B are independent, what is P(A | B)?
P(A)
Explain the concept of the expected value of a discrete random variable
The weighted average of all possible values of the random variable, with weights given by their respective probabilities.
Define the complement of an event in terms of probability.
The complement of an event A is the probability of not A, denoted as P(not A) or P(A').
Define the term "odds" in probability.
The ratio of the probability of success to the probability of failure.
In how many ways can you choose 3 items from a set of 8 distinct items?
C(8, 3)
How is the probability of the intersection of two independent events calculated?
P(A and B) = P(A) * P(B)
What is the probability mass function (PMF) of a discrete random variable?
A function that gives the probability of each possible outcome.
What is the difference between mutually exclusive and independent events?
Mutually exclusive events cannot occur together, while independent events are not influenced by each other.
Explain what a probability distribution is.
A table or formula that assigns probabilities to each possible outcome of a random variable.
What is the principle of inclusion-exclusion in combinatorics?
A counting principle used to correct for overcounting when calculating probabilities.
Explain what the multiplication rule is in terms of conditional probability.
P(A and B) = P(A | B) * P(B)
What is the central limit theorem and how does it relate to distributions?
The central limit theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed random variables approaches a normal distribution.
Explain what the Law of Large Numbers states in the context of probability.
As the number of trials increases, the experimental probability approaches the theoretical probability.
Define the term "expected value" in the context of probability.
The mean or average value of a random variable based on its probability distribution.
How many ways can you arrange the letters in the word "MISSISSIPPI"?
11! / (4! * 4! * 2!)
Define conditional probability and provide an example.
The probability of event A occurring given that event B has already occurred. Example: P(Rain | Cloudy).
Explain the difference between the mean and standard deviation of a probability distribution.
The mean is a measure of central tendency, while the standard deviation measures the spread or variability of the distribution.