Discrete Probability: every key term you need (+ practice quiz)
25 flashcard terms for Discrete Mathematics Topic 5, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
The set of all possible outcomes of an experiment. Every probability question begins by fixing this set, since the same informal question can have different answers under different sample spaces.
Event
Any subset of the sample space. Set operations translate directly into event language, with union meaning or, intersection meaning and, and complement meaning not.
Uniform probability model
A model in which every outcome of a finite sample space is equally likely, so the probability of an event is its size divided by the size of the sample space.
Probability axioms
The requirements that probabilities are nonnegative, that the whole sample space has probability one, and that disjoint events have probabilities that add.
Complement rule
The probability of an event not occurring equals one minus the probability that it occurs, which mirrors complementary counting and simplifies at least one problems.
Addition rule for probability
The probability of a union equals the sum of the individual probabilities minus the probability of the intersection, the probabilistic form of inclusion exclusion.
Conditional probability
The probability of an event given that another has occurred, defined as the probability of the intersection divided by the probability of the conditioning event, which must be nonzero.
Independent events
Two events are independent when the probability of their intersection equals the product of their probabilities, equivalently when conditioning on one leaves the other unchanged.
Mutually exclusive events
Events that cannot both occur, so their intersection is empty. Nonzero probability exclusive events are always dependent, since one occurring rules the other out entirely.
Multiplication rule for probability
The probability that two events both occur equals the probability of the first times the conditional probability of the second given the first, which chains naturally to more events.
Law of total probability
A rule that computes the probability of an event by splitting the sample space into a partition and summing the conditional probabilities weighted by the partition probabilities.
Bayes theorem
A formula that reverses a conditional probability, expressing the probability of a hypothesis given evidence in terms of the probability of the evidence given the hypothesis and the prior.
Prior and posterior probability
The prior is the belief in a hypothesis before evidence arrives and the posterior is the updated belief afterwards. Bayes theorem is the rule connecting them.
Base rate neglect
The common reasoning error of ignoring how rare a condition is when interpreting a test result, which is why an accurate test for a rare condition still yields many false positives.
Random variable
A function assigning a number to each outcome of a sample space. It is neither random nor a variable in the algebraic sense, but a rule for reading a numeric value off an outcome.
Probability distribution
The assignment of probabilities to the possible values of a random variable, with all values summing to one over the range of the variable.
Expected value
The probability weighted average of the values of a random variable, representing the long run mean over many repetitions rather than a value that must ever actually occur.
Linearity of expectation
The rule that the expectation of a sum equals the sum of expectations, which holds even when the summands are dependent and makes many hard counting problems easy.
Indicator random variable
A variable taking the value one when an event occurs and zero otherwise. Its expectation equals the probability of the event, which pairs powerfully with linearity of expectation.
Variance
The expected squared deviation from the mean, measuring spread. Its square root is the standard deviation, which is stated in the same units as the variable itself.
Bernoulli trial
A single experiment with exactly two outcomes, usually labelled success and failure, whose success probability stays fixed. Repeated independent trials generate the binomial setting.
Binomial distribution
The distribution of the number of successes in a fixed number of independent Bernoulli trials with constant success probability, with probabilities given by binomial coefficients.
Geometric distribution
The distribution of the number of independent trials until the first success, whose expected value is the reciprocal of the success probability.
Birthday problem
The classic result that only about twenty three people are needed for an even chance of a shared birthday, because the number of pairs grows quadratically with the group size.
Monte Carlo estimation
Approximating a quantity by repeated random sampling and averaging the results. Accuracy improves with the square root of the number of samples, so precision is expensive.