Statistics Ch. 5 - Probability
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)
Venn diagram
A diagram that uses circles contained within a rectangle to display elements of different sets. The rectangle represents the sample space, and circles represent events.
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.
probability model
lists the possible outcomes of a probability experiment and each outcome's probability
factorial symbol (n!)
if n ≥ 0 is an integer, the factorial symbol, n!, is defined as follows: n! = n(n-1)∗⋅⋅⋅∗3∗2∗1
addition rule for disjoint events
If E and F are disjoint events, then P(E or F) = P(E) +P(F)
multiplication rule of counting
If as task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, etc., then the task of making these selections can be done in p∗q∗r∗⋅⋅⋅ ways
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.
multiplication rule for n independent events
P(E and F and G and ...) = P(E) ∗ P(F) ∗P(G)
multiplication rule for independent events
P(E and F) = P(E) ∗ P(F)
general multiplication rule
P(E and F) = P(E) ∗ P(F|E)
general addition rule
P(E or F) = P(E) + P(F) - P(E and F)
equation for computing probability using the classical method
P(E) = (number of ways that E can occur)/ (number of possible outcomes) = m/n
equation for approximating probabilities using the empirical approach
P(E) ≈ relative frequency of E = (frequency of E)/(number of trials of experiment)
complement rule
P(Ec) = 1 - P(E)
number of permutations of distinct objects in groups
The number of arrangements of r objects chosen from n objects in which 1. the n objects are distinct 2. repetition of objects is not allowed 3. order is important nPr = n!/(n-r)!
number of combinations of n distinct objects taken r at a time
The number of different arrangements of n objects using r ≤ n of them, in which 1. the n objects are distinct 2. repetition of objects is not allowed 3. order is not important nCr = n! / [r!(n-r)!]
combination
a collection, without regard to order, of n distinct objects without repetition.
tree diagram
a diagram to determine a sample space that lists the equally likely outcomes of an experiment
fair die
a die where each possible outcome is equally likely
probability
a measure of the likelihood of a random phenomenon or chance behavior
subjective probability
a probability obtained on the basis of personal judgment
permutation
an arrangement in which r objects are chosen from n distinct objects, repetition is not allowed, and order is important.
unusual event
an event that has a low probability of occurring, typically less than 5%
impossible
an event with a probability of 0
certainty
an event with a probability of 1
simple event
an event with only one outcome
event
any collection of outcomes from a probability experiment, consisting of one or more outcomes
experiment
any process with uncertain results that can be repeated
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
nCr
combination of n objects taken r at a time
E
event
dependent events
events where the probability of one affects the probability of the other
independent events
events whose probability do not affect each other
n
number of equally likely outcomes
nPr
permutation of n objects taken r at a time
S
sample space
sample space
the collection of all possible outcomes
probability of an outcome
the long-term proportion with which a certain outcome is observed
m
the number of ways that an event E can occur
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
conditional probability
the probability that an event occurs, given that another event has occurred
mutually exclusive events
two events that have no outcomes in common; disjoint events
disjoint events
two events that have no outcomes in common; mutually exclusive events
