Probability

total probability, sometimes called the rule of elimination.

A generalization of the foregoing illustration to the case where the sample space is partitioned into k subsets is covered by the theorem of ?

discrete

A sample space is __________ if it consists of a finite or countable infinite set of outcomes.

continuous

A sample space is ____________ if it contains an interval (either finite or infinite) of real numbers.

If A and B are two events, then 𝑷 (𝑨 ∪ 𝑩) = 𝑷 (𝑨) + 𝑷 (𝑩) − 𝑷 (𝑨 ∩ 𝑩) .

Additive Rule or Theorem 2.7?

event

An ___________ is a subset of the sample space of a random experiment.

random experiment

An experiment that can result in different outcomes, even though it is repeated in the same manner every time, is called a _______________.

Bayesian statistics

- is a collection of tools that is used in a special form of statistical inference which applies in the analysis of experimental data in many practical situations in science and engineering.

combination

- is actually a partition with two cells, the one cell containing the r objects selected and the other cell containing the (𝒏 − 𝒓) objects that are left. The number of such combinations, denoted by 𝒏 𝒓, 𝒏 − 𝒓 , is usually shortened to 𝒏 𝒓 since the number of elements in the second cell must be 𝒏 − 𝒓.

cells

Often, we are concerned with the number of ways of partitioning a set of 𝑛 objects into 𝑟 subsets called ___________.

circular permutations

Permutations that occur by arranging objects in a circle are called ____________.

If an operation can be performed in n1 ways, and if each of these ways a second operation can be performed in n2 ways, then the two operations can be performed in n1n2 ways.

Rule 2.1?

If an operation can be performed in n1 ways, and if for each of these a second operation can be performed in n2 ways, and for each of the first two a third operation can be performed in n3 ways, and so forth, then the sequence of k operations can be performed in n1n2n3...nk ways.

Rule 2.2?

If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes corresponds to event A, then the probability of event A is 𝑷(𝑨) = 𝒏/𝑵

Rule 2.3:?

Pascal, Leibniz, Fermat, and James Bernoulli.

Some of the mathematicians providing these strategies for various games of chance:

multiplication rule

The fundamental principle of counting, often referred to as the _______________.

weights or probabilities

The likelihood of the occurrence of an event resulting from such a statistical experiment is evaluated by means of a set of real numbers, called ___________, ranging from 0 to 1.

conditional probability

The probability of an event B occurring when it is known that some event A has occurred is called a _____________ and is denoted by 𝑃(𝐵|𝐴). -is usually read "the probability that B occurs given that A occurs" or simply "the probability of B, given A".

Venn diagrams

The relationship between events and the corresponding sample space can be illustrated graphically by means of ________________.

sample space

The set of all possible outcomes of a random experiment is called the ____________ of the experiment. It is denoted as S.

The number of permutations of 𝑛 objects is 𝑛!.

Theorem 2.1:?

The number of permutations of 𝑛 distinct objects taken r at a time is the given formula nPr.

Theorem 2.2:?

The number of permutations of n objects arranged in a circle is (𝒏 − 𝟏)!.

Theorem 2.3:?

The number of distinct permutations of n things of which 𝒏𝟏 are of one kind, 𝒏𝟐 of a second kind, ..., 𝒏𝒌 of a kth kind is 𝒏!/(𝒏𝟏! 𝒏𝟐! ... 𝒏𝒌!)

Theorem 2.4: ?

The number of ways of partitioning a set of 𝒏 objects into 𝒓 cells with 𝒏𝟏 elements in the first cell, 𝒏𝟐 elements in the second, and so forth, is 𝒏 𝒏𝟏, 𝒏𝟐, ... , 𝒏𝒓 = 𝒏!/ (𝒏𝟏! 𝒏𝟐! ... 𝒏𝒓!) Where 𝒏𝟏 + 𝒏𝟐 + ⋯ + 𝒏𝒓 = 𝒏

Theorem 2.5:?

The number of combinations of n distinct objects taken r at a time is 𝒏 𝒓 = 𝒏!/ (𝒓! 𝒏 − 𝒓 !)

Theorem 2.6:?

For three events A, B, and C, 𝑷(𝑨 ∪ 𝑩 ∪ 𝑪) = 𝑷(𝑨) + 𝑷(𝑩) + 𝑷(𝑪) − 𝑷(𝑨 ∩ 𝑩) − 𝑷(𝑨 ∩ 𝑪) − 𝑷(𝑩 ∩ 𝑪) + 𝑷(𝑨 ∩ 𝑩 ∩ 𝑪).

Theorem 2.8:?

If A and A' are complementary events, then 𝑷(𝑨) + 𝑷(𝑨′) = 𝟏

Theorem 2.9:?

mutually exclusive

Two events E1 and E2 are ____________ , or disjoint, if E1 ∩ E2 = φ, that is, if E1 and E2 have no elements in common.

- Union - Intersection - Complement

basic set operations:

permutation

Definition 2.1 - is an arrangement of all or part of a set of objects.

n factorial

Definition 2.2: For any non-negative integer 𝑛, 𝑛!, called ____________, is defined as 𝒏! = 𝒏(𝒏 − 𝟏) · · · (𝟐)(𝟏), with special case 0! = 1

𝑷(𝑩|𝑨) = 𝑷(𝑨∩𝑩)/𝑷(𝑨) , provided 𝑷 𝑨 > 𝟎

Definition 2.4: The conditional probability of B, given A, denoted by 𝑃(𝐵|𝐴) , is defined by