BANA Exam 2
The range of probability is?
0-1
Relative Frequency Method
Assigning probabilities based on experimental or historical data
A method of assigning probabilities based upon judgment is referred to as the _____ method
Subjective
lamda=
average
Two events with nonzero probabilities
cannot be both mutually exclusive and independent
The standard deviation of a normal distribution
cannot be negative
If A and B are independent events with P(A) = 0.2 and P(B) = 0.6, then P(A union B) =
0.68
An experiment consists of three steps. There are four possible results on the first step, three possible results on the second step, and two possible results on the third step. The total number of experimental outcomes is?
24
The binomial probability distribution is used with a(n) _____ random variable
discrete
The probability distribution that can be described by just one parameter is the
exponential
There is a lower limit but no upper limit for a random variable that follows the _____ probability distribution
exponential
sample space
for an experiment is the set of all experimental outcomes
The _____ probability function is based in part on the counting rule for combinations
hypergeometric
To compute the probability that in a random sample of n elements, selected without replacement, we will obtain x successes, we would use the _____ probability distribution
hypergeometric
Random Experiment
is a process that generates well-defined experimental outcomes
standard deviation (σ)
is defined as the positive square root of the variance
discrete random variable
may assume either a finite number of values or an infinite sequence of values
The highest point of a normal curve occurs at
the mean
A negative value of z indicates
the number of standard deviations an observation is to the left of the mean
Bayes' theorem is used to compute
the posterior probabilities
The key difference between the binomial and hypergeometric distribution is that, with the hypergeometric distribution, the...
probability of success changes from trial to trial
Larger values of the standard deviation result in a normal curve that is
wider and flatter
A sample point refers to the?
Individual outcome of an experiment
A decision maker subjectively assigned the following probabilities to the four outcomes of an experiment: P(E1) = 0.15, P(E2) = 0.10, P(E3) = 0.45, P(E4) = 0.25. Are these probability assignments valid? Explain
No, the probabilities do not sum to 1
What is the complement of P(A|B)?
P(AC|B)
The counting rule that is used for counting the number of experimental outcomes when n objects are selected from a set of N objects where order of selection is important is called the counting rule for?
Permutations
When dealing with the number of occurrences of an event over a specified interval of time or space, the appropriate probability distribution is a _____ distribution
Poisson
Mutual Exclusiveness and Independence
• Do not confuse the notion of mutually exclusive events with that of independent events. • Two events with nonzero probabilities cannot be both mutually exclusive and independent. • If one mutually exclusive event is known to occur, the other cannot occur.; thus, the probability of the other event occurring is reduced to zero (and they are therefore dependent). Mutual Exclusiveness and Independence • Two events that are not mutually exclusive, might or might not be independent.
how to calculate in Uniform Probability Distribution
• Expected Value of x: E(x) = (a + b)/2 • Variance of x: Var(x) = (b - a)2/12
Independent Events
• If the probability of event A is not changed by the existence of event B, we would say that events A and B are independent. • Two events A and B are independent if: P(A|B) = P(A) or P(B|A) = P(B)
The sum of the probabilities of two complementary events is
1
Posterior probabilities are _____ probabilities
conditional
A normal probability distribution
-can have a mean of any numerical value -is a continuous probability distribution
Classical Method
Assigning probabilities based on the assumption of equally likely outcomes
Chapter 4 focuses on?
Probability
A graphical method of representing the sample points of an experiment is a?
Tree diagram
If A and B are mutually exclusive events with P(A) = 0.295, P(B) = 0.32, then P(A | B) =
0.0000
An experiment consists of four outcomes with P(E 1) = 0.2, P(E 2) = 0.3, and P(E 3) = 0.4. The probability of outcome E 4 is
0.1000
In a standard normal distribution, what is the probability that z is greater than zero?
0.5
Two Properties of a Poisson Experiment
1. The probability of an occurrence is the same for any two intervals of equal length 2. The occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval
Which of the following is not a characteristic of the normal probability distribution?
99.72% of the time the random variable assumes a value within plus or minus 1 standard deviation of its mean
The union of events A and B is the event containing all the sample points belonging to?
A or B or both
Subjective Method
Assigning probability based on judgment
The counting rule that is used for counting the number of experimental outcomes when n objects are selected from a set of N objects where order of selection is not important is called the rule for?
Combinations
What is a collection of sample points called?
Event
In the textile industry, a manufacturer is interested in the number of blemishes or flaws occurring in each 100 feet of material. The probability distribution that has the greatest chance of applying to this situation is the
Poisson distribution
Initial estimates of the probabilities of events are known as _____ probabilities
Prior
When the results of experimentation or historical data are used to assign probability values, the method used to assign probabilities is referred to as the _____ method
Relative frequency
An experimental outcome is also called a
Sample point
The symbol ∪ shows the?
Union of events
A graphical representation in which the sample space is represented by a rectangle and events are represented as circles is called a?
Venn diagram
The Poisson probability distribution is used with _____ random variable
a discrete
The number of customers that enter a store during one day is an example of
a discrete random variable
A normal distribution with a mean of 0 and a standard deviation of 1 is called
a standard normal distribution
Assume you have applied for two jobs—Job A and Job B. The probability that you get an offer for only Job A is 0.22. The probability of being offered only Job B is 0.18. The probability of getting at least one of the jobs is 0.43. a. What is the probability that you will be offered both jobs? b. Are Events A and B mutually exclusive? Why or why not? Explain
a. 0.03 b. No, because P(A ∩ B) ≠ 0.
Assume you have applied for two scholarships, a Merit scholarship (M) and an Athletic scholarship (A). The probability that you receive an Athletic scholarship is 0.22. The probability of receiving both scholarships is 0.14. The probability of getting at least one of the scholarships is 0.35. a. What is the probability that you will receive a Merit scholarship? b. Are Events A and M mutually exclusive? Why or why not? Explain. c. Are Events A and M independent? Explain using probabilities.
a. 0.27 b. No, because P(A ∩ M) ≠ 0. c. No, because P(A ∩ M) ≠ P(A)P(M).
A probability distribution showing the probability of x successes in n trials, where the probability of success does not change from trial to trial, is termed a
binomial probability distribution
The z score for the standard normal distribution
can be either negative or positive
A measure of the average value of a random variable is called a(n)
expected value
A continuous probability distribution that is useful in describing the time, or space, between occurrences of an event is a(n) _____ probability distribution
exponential
The symbol ∩ shows the?
intersection of two events
random variable
is a numerical description of the outcome of an experiment
Probability
is a numerical measure of the likelihood that an event will occur. • Probability values are always assigned on a scale from 0 to 1. • A probability near zero indicates an event is quite unlikely to occur. d A probability near one indicates an event is almost certain to occur
The intersection of two mutually exclusive events
must always be equal to 0
expected value
or mean, of a random variable is a measure of its central location
The function that defines the probability distribution of a continuous random variable is a
probability density function
Chapter 5 focuses on
probability distributions
A numerical description of the outcome of an experiment is called a
random variable
If two events are independent, then
the product of their probabilities gives their intersection
Normal Probability Distribution: • Characteristics (basis for the empirical rule)
• 68.26% of values of a normal random variable are within +/- 1 standard deviation of its mean • 95.44% of values of a normal random variable are within +/- 2 standard deviations of its mean • 99.72% of values of a normal random variable are within +/- 3 standard deviations of its mean
Uniform Probability Distribution
• A random variable is uniformly distributed whenever the probability is proportional to the interval's length • The uniform probability density function is: f (x) = 1/(b - a) for a < x < b = 0 where: a = smallest value the variable can assume b = largest value the variable can assume
Standard Normal Probability Distribution
• Converting to the Standard Normal Distribution z = (𝑥−𝜇) / 𝜎 -If it is to the right you have to do 1 - ((𝑥−𝜇) / 𝜎)
Exponential Probability Distribution
• Density Function where: 𝑓(𝑥) = 1/𝜇 𝑒−𝑥/𝜇 for𝑥> 0 µ = expected value or mean e = 2.71828 OR 𝑓(𝑥) = 𝜆𝑒−𝜆𝑥 for 𝑥> 0 where: λ = 1/µ e = 2.71828
Using Excel to Compute Normal Probabilities • Excel has two functions for computing cumulative probabilities and z values for a standard normal distribution:
• NORM.S.DIST is used to compute the cumulative probability given an z value • NORM.S.INV is used to compute the z value given a cumulative probability
Addition Law
• The addition law provides a way to compute the probability of event A, or B, or both A and B occurring • The law is written as: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Multiplication Law for Independent Events
• The multiplication law also can be used as a test to see if two events are independent • The law is written as: P(A ∩ B) = P(A)P(B)
Multiplication Law
• The multiplication law provides a way to compute the probability of the intersection of two events • The law is written as: P(A ∩ B) = P(B)P(A|B) or P(A ∩ B) = P(A)P(B|A)
Conditional Probability
• The probability of an event given that another event has occurred is called a conditional probability • The conditional probability of A given B has already occurred is denoted by P(A|B) • A conditional probability is computed as follows : 𝑃(𝐴/𝐵) = 𝑃(𝐴∩𝐵) / 𝑃(𝐵)
A random variable that can assume only a finite number of values is referred to as a(n)
discrete random variable
Using Excel to Compute Normal Probabilities • Excel has two functions for computing cumulative probabilities and x values for any normal distribution:
• NORM.DIST is used to compute the cumulative probability given an x value • NORM.INV is used to compute the x value given a cumulative probability
variance
•Summarizes the variability in the values of a random variable •The variance is a weighted average of the squared deviations of a random variable from its mean. The weights are the probabilities
Complement of an Event
•The complement of event A is defined to be the event consisting of all sample points that are not in A •The complement of A is denoted by Ac
Intersection of Two Events
•The intersection of events A and B is the set of all sample points that are in both A and B •The intersection of events A and B is denoted by A ∩ B.
Union of Two Events
•The union of events A and B is the event containing all sample points that are in A or B or both •The union of events A and B is denoted by A ∪ B
When the assumption of equally likely outcomes is used to assign probability values, the method used to assign probabilities is referred to as the _____ method
Classical
Which of the following is a characteristic of an experiment where the binomial probability distribution is applicable?
Exactly two outcomes are possible on each trial
The probability of an event is the _____ of the probabilities of the sample points in the event
Sum
The collection of all possible sample points in an experiment is?
The sample space
In a binomial experiment, the probability
does not change from trial to trial
The probability that a continuous random variable takes any specific value
is equal to zero
The center of a normal curve
is the mean of the distribution
Normal Probability Distribution
is the most important distribution for describing a continuous random variable
continuous random variable
may assume any numerical value in an interval or collection of intervals
The addition law is potentially helpful when we are interested in computing the probability of?
the union of two events
Poisson Probability Distribution
• A Poisson distributed random variable is often useful in estimating the number of occurrences over a specified interval of time or space. • It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2, . . )
Binomial Probability Distribution
• Four Properties of a Binomial Experiment 1. The experiment consists of a sequence of n identical trials 2. Two outcomes, success and failure, are possible on each trial 3. The probability of a success, denoted by p, does not change from trial to trial. (This is referred to as the stationarity assumption.) 4. The trials are independent
Mutually Exclusive Events
• Two events are said to be mutually exclusive if the events have no sample points in common • Two events are mutually exclusive if, when one event occurs, the other cannot occur • If events A and B are mutually exclusive, P(A ∩ B) = 0 • The addition law for mutually exclusive events is: P(A ∪ B) = P(A) + P(B)
Hypergeometric Probability Distribution
• the trials are not independent • the probability of success changes from trial to trial
