Statistics-Q381

Multiplication Rule for the conditional probability of B

The conditional probability of event B occurring given that event A has occurred is P(B|A) = P(A and B) / P(A).

permutation of n objects taken r at a time

nPr = n! / (n-r)! where r is <= n

P(A or B or C)

= P(A) + P(B) + P(C) - P(A and B) - P(B and C) - P(A and C) + P(A and B and C)

combinations of n objects taken r at a time

A combination is a selection of r objects from a group of n objects without regard to order and is denoted by nCr. nCr = n!/((n-r)!r!)

event

A subset of the sample space. It may consist of one or more outcomes.

random

by chance

Mutually exclusive events

Two events A and B that cannot occur at the same time.

standard deviation of a discrete random variable

sigma = square root of the variance of a discrete random variable

variance of a binomial distribution

sigma squared = npq

variance of a discrete random variable

sigma squared = sum [(x-mu)squared times P(x)] Shortcut formula = sigma squared = [sum of x squared times P(x)] - mu squared

standard deviation of a binomial distribution

sqrt(npq)

classical (theoretical) probability

used when each outcome in a sample space is equally likely to occur. The classical probability for an event E is given by P(E) = (number of outcomes in event E) divided by (total number of outcomes in sample space).

distinguishable permutations of n objects

where n1 are of one type, n2 of another type and so on is, n! / (n1!)(n2!)(n3!) . . . (nk!) where n1 + n2 + n3+ . . . +nk = n.

steps to construct a discrete probability distribution

1. Make a frequency distribution for the possible outcomes. 2. Find the sum of the frequencies. 3. Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies. 4. Check that each probability is between 0 and 1, and that the sum of all the probabilities is 1.

subjective probability

results from intuition, educated guesses, and estimates.

expected value of a discrete random variable

= mean of the random variable. = E(x) = mu = sum(x times P(x)).

binomial experiment

A probability experiment that satisfies the following conditions: (1) The experiment is repeated for a fixed number of trials (n), where each trial is independent of the other trials, (2) There are only 2 possible outcomes of interest for each trial -- success (S) or failure (F), (3) The probability of a success, P(S) or p, is the same for each trial, (4) The random variable x counts the number of successful trials. Note: P(F)or q = probability of failure. q = 1-p

tree diagram

A visual display of the outcomes of a probability experiment by using branches that originate from a starting point. It can be used to find the number of possible outcomes in a sample space as well as individual outcomes.

probability experiment

An action, or trial, through which specific results (counts, measurements, or responses) are obtained.

simple event

An event that consists of a single outcome.

permutation

An ordered arrangement of objects. The number of different permutations of n objects is n!.

Law of Large Numbers

As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event.

empirical (statistical) probability

Based on observations obtained from probability experiments. The empirical probability of an event E is the relative frequency of an event. P(E) = (frequency of event E) divided by (total frequency) = f divided by n

Fundamental Counting Principle

If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m times n. This rule can be extended to any number of events occurring in sequence.

independent/dependent events

If the occurrence of one of the events does not affect the probability of the occurrence of the other event. Two events A and B are independent if P(B|A) = P(B) or if P(A|B) = P(A). Events that are not independent are dependent.

Bayes' Theorem

P(A|B) = (P(A) times P(B|A)) / (P(A) times P(B|A) + P(A') times P(B|A'))

binomial probability formula

P(x) = nCx(p^x)(q^(n-x)) = (n!/((n-x)!x!))(p^x)(q^(n-x))

odds of winning

Ratio of the number of successful outcomes to the number of unsuccessful outcomes

odds of losing

Ratio of the number of unsuccessful outcomes to the number of successful outcomes

Range of Probabilities Rule

The probability of an event E is between 0 and 1, inclusive. That is, 0 is less than or equal to P(E) and P(E) is less than or equal to 1. 1 = certain, 0 = impossible, 0.5 = even chance, <0,05 = unusual

conditional probability

The probability of an event occurring, given that another event has already occurred. The conditional probability of event B occurring, given that A has occurred, is denoted by P(B|A) and is read as "probability of B, given A".

Addition Rule for the probability of A or B

The probability that events A or B will occur, P(A or B), is given by P(A or B) = P(A) + P(B) - P(A and B). If events A and B are mutually exclusive, then the rule can be simplified to P(A or B) = P(A) + P(B). This simplified rule can be extended to any number of mutually exclusive events.

Multiplication Rule for the probability of A and B

The probability that two events A and B will occur in sequence is P(A and B) = P(A) times P(B|A). If events A and B are independent, then the rule can be simplified to P(A and B) = P(A) times P(B). This simplified rule can be extended to any number of independent events.

outcome

The result of a single trial in a probability experiment.

complement of event E

The set of all outcomes in a sample space, that are not included in event E. The complement of event E is denoted by E' and is read as "E prime". P(E) + P(E') = 1

sample space

The set of all possible outcomes of a probability experiment.

Poisson distribution

a discrete probability distribution of a random variable x that satisfies the following conditions: (1) The experiement consists of counting the number of times x an event occurs in a given interval. The interval can be an interval of time, area, or volume. (2) The probability of the event occurring is the same for each interval, (3) the number of occurrences in one interval is independent of the number of occurrences in other intervals. The probability of exactly x occurrences in an interval is P(x) = (mu^x times e^(-mu))/x! where e is an irrational number approximately equal to 2.71828 and mu is the mean number of occurrences per interval unit.

geometric distribution

discrete probability distribution of a random variable x that satisfies the following conditions: (1) a trial is repeated until a success occurs, (2) the repeated trials are independent of each other, (3) the probability of success p is constant for each trial, (4) the random variable x represents the number of the trial in which the first success occurs. The probability that the first success will occur on trial number x is P(x) = pq^(x-1), where q = 1-p.

discrete random variable

has a finite or countable number of possible outcomes that can be listed.

continuous random variable

has an uncountable number of possible outcomes, represented by an interval on the number line

discrete probability distribution

lists each possible value the random variable can assume, together with its probability. Satisfies the following conditions: (1) probability of each value of the discrete random variable is between 0 and 1, inclusive, (2) the sum of all probabilities is 1.

mean of a binomial distribution

mu = np

mean of a discrete random variable

mu = sum(x times P(x)). Each value of x is multiplied by its corresponding probability and the products are added.

random variable

represents a numerical value (count or measure)associated with each outcome of a probability experiment.