Probability
77 flashcards by MentisLibrary
Counting rules, conditional probability, Bayes, expected value, and the distributions that model real processes. Includes the classic paradoxes that break most people's intuition.
- What is a sample space?
- The set of all possible outcomes of a random experiment
- What is an event?
- Any subset of the sample space
- What are the three axioms of probability?
- Probabilities are non-negative, the whole sample space has probability 1, and disjoint events add
- What is the complement rule?
- The probability an event does not happen is one minus the probability it does
- What is the general addition rule?
- Add the individual probabilities and subtract the overlap
- What does it mean for events to be mutually exclusive?
- They cannot both happen, so their intersection is empty
- What does it mean for events to be independent?
- Knowing one occurred does not change the probability of the other
- What is the general multiplication rule?
- The probability both occur is the probability of the first times the conditional probability of the second
- What is conditional probability?
- The probability of $A$ given that $B$ has occurred
- Why is $P(A|B)$ generally not equal to $P(B|A)$?
- Because the two conditions restrict different sample spaces
- What is Bayes' theorem?
- A rule for updating a probability after new evidence
- What is the law of total probability?
- Split the sample space into cases and add the weighted probabilities
- What is the base rate fallacy?
- Ignoring how rare a condition is when interpreting a positive test
- What is the fundamental counting principle?
- If one stage has $m$ options and the next has $n$, together they have $mn$
- What is a permutation?
- An arrangement where order matters
- What is a combination?
- A selection where order does not matter
- How do you count arrangements with repeated items?
- Divide the total factorial by the factorial of each repeat count
- How does sampling with replacement differ from without?
- With replacement the trials are independent and the pool stays the same, without replacement each draw changes the pool
- What is the inclusion-exclusion principle for three events?
- Add the singles, subtract the pairs, add back the triple
- What are odds and how do they relate to probability?
- Odds in favor are the ratio of success to failure chances
- What is a random variable?
- A function assigning a number to each outcome of a random experiment
- What is expected value?
- The long-run average value of a random variable
- What is the linearity of expectation?
- Expectation of a sum is the sum of expectations, whether or not the variables are independent
- How do you compute the variance of a random variable?
- The expected squared deviation from the mean
- How do scaling and shifting affect mean and variance?
- Shifting moves the mean but not the variance, scaling multiplies the mean by $a$ and the variance by $a^2$
- When does $\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y)$?
- When $X$ and $Y$ are independent, or more generally uncorrelated
- What is covariance?
- A measure of how two variables move together
- What is a Bernoulli trial?
- A single experiment with exactly two outcomes, success with probability $p$ and failure with $1-p$
- What is the binomial distribution?
- The number of successes in $n$ independent trials each with probability $p$
- What are the mean and variance of a binomial distribution?
- Mean $np$ and variance $np(1-p)$