Elementary Statistics Chapter 4

Conditional probability

The probability that an event will occur given that another has already occurred. If A and B are two events, then the conditional probability of A given B is written as: P(A I B), and read as "the probability of A given that B has already occurred."

Classical Probability

Two or more outcomes that have the same probability of occurrence are said to be equally likely outcomes.

Compound event

A collection of more than one outcome for an experiment.

Event

A collection of one or more of the outcomes of an experiment.

Probability

A numerical measure of the likelihood that a specific event will occur.

Experiment

A process that, when performed, results in one and only one of many observations.

Simple event

An event that includes one and only one of the (final) outcomes for an experiment is called a simple event and is denoted by Ei

Combinations

Combinations give the number of ways x elements can be selected from n elements. The notation used to denote the total number of combinations is: nCx Which is read as "the number of combinations of n elements selected x at a time."

Mutually exclusive events

Events that cannot occur together

Calculating conditional probability

If A and B are two events, then, P(B I A)=P(A and B)/P(A) and P(A I B)=P(A and B)/P(B) given that P (A) ≠ 0 and P (B ) ≠ 0.

Counting Rule: To find total outcomes

If an experiment consists of three steps and if the first step can result in m outcomes, the second stem in n outcomes, and the third in k outcomes, then: Total outcomes for the experiment=m(n)(k)

Law of large numbers

If an experiment is repeated again and again, the probability of an event obtained from the relative frequency approaches the actual (or theoretical) probability.

Relative frequency concept of probability

If an experiment is repeated n times and an event A is observed f times where f is the frequency, then, according to the relative frequency concept of probability: P(A)=f/n=Frequency of A/Sample size

Simulating the tosses of a coin

If the experiment is repeated only a few times, the probabilities obtained may not be close to the actual probabilities. As the number of repetitions increases, the probabilities of out comes obtained become very close to the actual probabilities.

Intersection of events

Let A and B be two events defined in a sample space. The intersection of A and B represents the collection of all outcomes that are common to both A and B and is denoted be (A and B)

Union of events

Let A and b be two events defined in a sample space. The union of events A and B is the collection of all outcomes that belong to either A or B or to both A and B and is denoted by: (A or B)

Classical Probability Rule to Find Probability

P(Ei)=1/Total number of outcomes for the experiment P(A)=Number of outcomes favorable to A/Total number of outcomes for the experiment

Probability of obtaining a head and the probability of obtaining a tail for one toss of a coin.

P(head)=1/Total number of outcomes=1/2=.50 Similarly, P(tail)=1/2=.50

Compute the conditional probability P(in favor I male) for the data on 100 employees.

P(in favor I male)=Number of males who are in favor/Total number of males=15/60=0.25

Permutations notation

Permutations give the total selections of x element from n (different) elements in such a way that the order of selections is important. The notation used to denote the permutations is: nPx Which is read as "the number of permutations of selecting x elements from n element." Permutations are also called ARRANGEMENTS.

Sample space

The collection of all outcomes for an experiment is called a sample space.

Complement of event A

The complement of event A, denoted by Ā and read as "A bar" or "A complement," is the event that includes all the outcomes for an experiment that are not in A.

Permutations formula

The following formula is used to find the number of permutations or arrangements of selecting x items out of n items. Note that here, the n items must all be different. nPx=n!/(n-x)!

Joint probability of mutually exclusive events

The joint probability of two mutually exclusive events is always zero. If A and B are two mutually exclusive events, then: P(A and B)=0

Number of combinations

The number of combinations for selecting x from n distinct elements is given by the formula: nCx=n!/x!(n-x!) Where n!, x!, and (n-x)! are read as "n factorial," "x factorial," "n minus x factorial," respectively.

Addition Rule: To find the probability of union of two mutually nonexclusive events

The portability of the union of two mutually nonexclusive events A and B is: P(A or B)=P(A)+P(B)-P(A and B)

Subjective probability

The probability assigned to an event based on subjective judgement, experience, information, and belief. There are no definite rules to assign such probabilities.

Marginal probability

The probability of a single event without consideration of any other event.

First property of probability

The probability of an event always lies in the range 0 to 1.

Multiplication rule to find joint probability of two dependent events

The probability of the intersection of two dependent events A and B is: P(A and B) = P(A) x P(B I A) OR P(B) x P(A I B)

Multiplication rule: joint probability

The probability of the intersection of two events is called their joint probability. It is written as P(A and B).

Multiplication rule to calculate the probability of independent events

The probability of the intersection of two independent events A and B is: P(A and B) = P(A) x P(B)

Addition Rule: To find the probability of the union of mutually exclusive events

The probability of the union of two mutually exclusive events A and B is: P(A or B) = P(A) + P(B)

Second property of probability

The sum of the probabilities of all simple events (or final outcomes) for an experiment, denoted by ΣP(Ei), is always 1.

Factorials

The symbol n!, read as "n factorial," represents the product of all the integers from n to 1. In other words: n! = n(n - 1)(n - 2)(n - 3) · · · 3 · 2 · 1 By definition, 0! = 1

Outcomes

These observations are called the outcomes of the experiment.

Independent events

Two events are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other. ex: either: P(A I B)=P(A) OR P(B I A)=P(B)

nCx

n: denotes the total number of elements nCx= the number of combinations of n elements selected x at a time x: denotes the number of elements selected per selection.