Chapter 5

A fair die is rolled 10 times. What is the probability that an even number (2, 4, or 6) will occur between 2 and 4 times?

.3662 Σ P(X) = n!/[x!(n − x)!] × px(1 − p)n−x, for x = 2, 4, 6; or look up in binomial table, where x = 2 or 4 or 6 when p = .3 and n = 10.

discrete probability distribution

a table, graph or formula that gives the probability associated with each possible value that the variable can assume Denote the values of the random variable by x and the value's associated probability by p(x)

Random Variable

a variable whose value is a numerical value that is determined by the outcome of an experiment

If x is a binomial random variable, then the standard deviation of x is given by

√npq

In a binomial distribution, the random variable X is continuous

False The binomial distribution defines the random variable as discrete

The number of ways to arrange x successes among n trials is equal to

(n!)/(x!(n-x)!

A fair die is rolled 10 times. What is the probability that an odd number (1, 3, or 5) will occur fewer than 3 times?

.0547 Σ P(X) = n!/[x!(n − x)!] × px(1 − p)n−x, for x = 1, 3, 5; or look up in binomial table where x = 1 or 3 or 5 when p = .3 and n = 10.

Binomial Experiment characteristics

1). Experiment consists of n identical trials 2). each trial results in either "success" or "failure" 3). Probability of success, p, is constant from trial to trial the probability of failure, q, is 1-p 4). trials are independent If x is the total number of successes in n trials of a binomial experiment, then x is a binomial random variable

discrete probability distribution properties

1. For any value x of the random variable 2. The probabilities of all the events in the sample space must sum to 1

The requirement that the probability of success remains constant from trial to trial is a property of the _________________ distribution

Binomial

If p = .5 and n = 4, then the corresponding binomial distribution is

Symmetric

Expected Value of a Discrete Random Variable

The mean or expected value of a discrete random variable X is

variance of a discrete random variable standard deviation of a discrete random variable

The variance is the average of the squared deviations of the different values of the random variable from the expected value The standard deviation is the square root of the variance The variance and standard deviation measure the spread of the values of the random variable from their expected value

The binomial experiment consists of n independent, identical trials, each of which results in either success or failure and is such that the probability of success on any trial is the same.

True

The internal auditor for your company believes that 10 percent of their invoices contain errors. To check this theory, 20 invoices are randomly selected and 5 are found to have errors. The claim of the auditor will be rejected.

True Reject claim because P < .05.

q=

q=1-p

Which of the following is not a discrete random variable? a). the number of minutes required to run 1 mile b). the number of times a light changes red in a 10-minute cycle c). the number of criminals found in a five-mile radius of a neighborhood d). the number of defects in a sample selected from a population of 100 products

a). the number of minutes required to run 1 mile

The mean of the binomial distribution is equal to

np

the equation for the variance of the binomial distribution is given by

np(1 − p).