Relative Frequency - Probability of an event is proportion of times the event occurs in a long series of repetitions of an experiment / process.
E.g probability of getting a 2 from a rolled dice is
Subjective Probability - Probability of an event is a measure of how sure person making the statement is that the event will happen
Discrete - Consists of a finite or countably infinite set of sample points
Simple definition
Let be a discrete sample space
Odds in favor of an event A is the probability the event occurs divided by the probability it does not occur, or simply , odds against the event is reciprocal of this or simply
Additional Rule - Do job 1 in p ways and job 2 in q ways, then we can do either job 1 OR job 2 (but not both) in p + q ways
Multiplication Rule Do job 1 in p ways and for each of these ways we can do job 2 in q ways. Then we can do both job 1 AND job 2 in p x q ways
ordered arrangements of length n using each symbol once and only once. This is denoted by
Permutation - ordered arrangements of length each symbol at most once. This product is denoted by (read “n to k factors”). Note that (ORDER DOES MATTER)
Combination - (ORDER DOES NOT MATTER)
Inclusion Exclusion Principle -
3 events:
Conditional Probability - A and B be two events such that then
Independence -
Law of Total Probability - where if then for any event, A,
Probability Mass Function - Mapping each outcome to the probability of it happening
Cumulative Distribution Function - Cumulatively adding up the outcomes of the previous event
Binomial Distribution - n independent trials with 2 possible outcomes
success: probability p
failure: probability 1 - p
X = # of successes in n trials then X follows a Binomial Distribution with params n and p
X ~ Bin(n, p)
Geometric Distribution - X is number of trials required to observe first success in a sequence of experiments
X ~ Geo(p) then E(X) = and Var(X) =
Negative Binomial Distribution - X ~ NB(r, p)
X is the number of trials required to observe the success