probability
a measure of the likelihood of a random phenomenon or chance behavior
probability of an outcome
the long-term proportion with which a certain outcome is observed
the law of large numbers
as the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome
experiment
any process with uncertain results that can be repeated
sample space
the collection of all possible outcomes. S
event
any collection of outcomes from a probability experiment, consisting of one or more outcomes. E
simple event
an event with only one outcome
probability model
lists the possible outcomes of a probability experiment and each outcome's probability
impossible
an event with a probability of 0
certainty
an event with a probability of 1
unusual event
an event that has a low probability of occurring, typically less than 5%
equation for approximating probabilities using the empirical approach
P(E) ? relative frequency of E =
(frequency of E)/(number of trials of experiment)
equation for computing probability using the classical method
P(E) = (number of ways that E can occur)/ (number of possible outcomes) = m/n
n
number of equally likely outcomes
m
the number of ways that an event E can occur
tree diagram
a diagram to determine a sample space that lists the equally likely outcomes of an experiment
subjective probability
a probability obtained on the basis of personal judgment
disjoint events
two events that have no outcomes in common; mutually exclusive events
mutually exclusive events
two events that have no outcomes in common; disjoint events
addition rule for disjoint events
If E and F are disjoint events, then P(E or F) = P(E) +P(F)
Benford's Law
Mathematical algorithm that accurately predicts that, for many data sets, the first digit of each group of numbers in a random sample will begin with 1 more than a 2, a 2 more than a 3, a 3 more than a 4, and so on. Predicts the percentage of time each digit will appear in a sequence of numbers.
general addition rule
P(E or F) = P(E) + P(F) - P(E and F)
contingency table
A table that relates two categories of data; two-way table. Variables are placed in rows and columns; each intersection of variables is a cell in the table.
complement of an event
the probability that an event does not occur; all outcomes in a sample space that are not outcomes in the event
complement rule
P(Ec) = 1 - P(E)
independent events
events whose probability do not affect each other
dependent events
events where the probability of one affects the probability of the other
multiplication rule for independent events
P(E and F) = P(E) ? P(F)
multiplication rule for n independent events
P(E and F and G and ...) = P(E) ? P(F) ?P(G)
Rules of probability
1. The probability of any event must be between 0 and 1, inclusive. 0 ? P(E) ? 1.
2. The sum of the probabilities of all outcomes must equal 1.
3. If E and F are disjoint events, then P(E or F) = P(E) + P(F). If E and F are not disjoint events, then P(E or F) = P(E) + P(F) - P(E and F)
4. If E represents any event and Ec represents the complement of E, then P(Ec) = 1 - P(E)
5. If E and F are independent events, then P(E and F) = P(E)?P(F)
conditional probability
the probability that an event occurs, given that another event has occurred
general multiplication rule
P(E and F) = P(E) ? P(F|E)