Business Statistics Chapter 5
In the venn diagram, what is the joint probability?
The overlap is P(A and B)
EVENT
A collection of one or more outcomes of an experiment.
given an experiment of rolling a die and the event getting an eve, which of the following event would make the set of events collectively exhaustive events?
getting an odd
Classical Probability
is based on the assumption that the outcomes of an experiment are equally likely Probability of an event = Number of favorable outcomes/Total number of possible outcomes
a bag contains 10 marbles: 4 red, 4 white, and 2 blue. what is the probability of drawing a blue marble?
.2 2/10
The probability of rolling a one with a die is 1/6. what is the probability of rolling two ones with two dice?
1/36 P(A and B) = P(A)P(B)
A fair coin (that is, the probability of "heads" equals 50%) is repeated flipped. What does the law of large numbers predict?
As the number of flips increases, the proportion of "heads" will approach 1/2
Which of the following are true regarding the special rule of addition and the general rule of addition? select all that apply
If the events A and B are not mutually exclusive, you an use the general rule of addition. if the events A and B are mutually exclusive, you can use the special rule of addition
LAW OF LARGE NUMBERS
Over a large number of trials, the empirical probability of an event will approach its true probability.
Which of the following formulas correctly represents the general rule of multiplication for two events?
P(A and B) = P(A)P(B|A)
Which one of the following conditions must be met to use the special rule of addition? i.e. P(A or B) = P(A) + P(B)
The events A an B must be mutually exclusive
In the Venn Diagram, the area where the circles overlap is P(A and B). What is this area called?
The joint probability of A and B
INDEPENDENCE
The occurrence of one event has no effect on the probability of the occurrence of another event.
How would you read the formula P(~A)?
The probability of Not A
tree diagram
is a visual that is helpful in organizing and calculating probabilities for problems, similar to the previous example/solution.
Study the venn diagram shown. what is the value of P(E or F)? P(E) = .40 P(F) = .30 P(D) = .10
.60 P(E or F) = .40 + .30 - .10
If P(J)=0.4 and P(Z)=0.3, what is the probability that J and Z occur together? assume the events are independent
0.12 (.4)(.3)=.12
Given the following contingency table, how many conditional probabilities are listed for the second set of branches? Assume 'Movies Attended' is represented as the first stage of the tree diagram.
6 - for each category of 'movies attended,' there are two branches (one for each gender)
Assume a die is rolled. For which one of the following sets of events could the special rule of addition be used?
A = getting a 2 B = getting a 6 Because they are mutually exclusive
OUTCOME
A particular result of an experiment.
JOINT PROBABILITY
A probability that measures the likelihood two or more events will happen concurrently
Choose the statement that best defines the term "experiment" in the context of probability.
A process that leads to only one of several possible outcomes.
Experiment
A process that leads to the occurrence of one and only one of several possible results
Special rule of addition
A rule used to find the probabilities of events made up of A or B when the events are and must be, mutually exclusive. P(A or B)=P(A)+P(B)
CONTINGENCY TABLE
A table used to classify sample observations according to two or more identifiable characteristics. A contingency table is a cross-tabulation that simultaneously summarizes two variables of interest and their relationship.
Venn Diagram
English logician J. Venn (1834-1923) developed a diagram to portray graphically the outcome of an experiment. The mutually exclusive concept and various other rules for combining probabilities can be illustrated using this device. To construct a Venn diagram, a space is first enclosed representing the total of all possible outcomes. This space is usually in the form of a rectangle. An event is then represented by a circular area that is drawn inside the rectangle proportional to the probability of the event. The following Venn diagram represents the mutually exclusive concept. There is no overlapping of events, meaning that the events are mutually exclusive. In the following Venn diagram, assume the events A, B, and C are about equally likely
Considering the experiment of rolling a die, which event would be both mutually exclusive and collectively exhaustive with the event getting a number greater than 4?
Getting a number less than 5
SPECIAL RULE OF MULTIPLICATION
P(A and B)=P(A)P(B) For two independent events A and B, the probability that A and B will both occur is found by multiplying the two probabilities
SUBJECTIVE CONCEPT OF PROBABILITY
The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available.
MUTUALLY EXCLUSIVE
The occurrence of one event means that none of the other events can occur at the same time
What is the purpose of using selection with replacement?
The probabilities do not change from one trial to another, so events are independent. each experiment uses the same set of events.
CONDITIONAL PROBABILITY
The probability of a particular event occurring, given that another event has occurred.
EMPIRICAL PROBABILITY
The probability of an event happening is the fraction of the time similar events happened in the past. Empirical probability = Number of times the event occurs/Total number of observations
What is empirical probability?
The relative frequency with which the event happened in the past. Empirical probability is based on history.
if the fraction of times an event happened in the past is used as the basis for assigning a probability to the event, this is _________ probability
empirical
a dice cube is tossed and the number of spots on the uppermost face is noted. match the term to the part of the random trial to which is refers
experiment -> the process of tossing the die outcome -> the die is a two event -> the die is greater than two
in a probability experiment involving rolling a die, which of the following is an outcome? select all that apply getting a 2 getting a number less than 4 getting a 6 getting an even
getting a 2 getting a 6 getting one out of many possible outcomes
The event ~A
is the complement of event A
The probability that event A will occur is forty-five percent. Which of these statements correctly expresses this statement in mathematical form?
P(A) = 0.45
Which of the following is a conditional probability?
The chance an employee will do well on the job given the employee scored well on a pre-employment test
PROBABILITY
A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur.
Collectively Exhaustive
At least one of the events must occur when an experiment is conducted. If an experiment has a set of events that includes every possible outcome, such as the events "an even number" and "an odd number" in the die-tossing experiment. For the die-tossing experiment, every outcome will be either even or odd. So the set is collectively exhaustive. If the set of events is collectively exhaustive and the events are mutually exclusive, the sum of the probabilities is 1
suppose 6 students are randomly selected and asked if they live on campus. which of the following is a possible event of this experiment? select all that apply
less than half say they live on campus all six say they live on campus none say they live on campus
which of the following statements accurately repeats the formula P(B)=0.30?
the probability that the event "B" will occur is thirty percent
GENERAL RULE OF ADDITION
used to compute the probability of two events that are NOT mutually exclusive P(A or B)=P(A)+P(B)−P(A and B)
Select the methodology that would result in a subjective probability
weighing the available information and assigning a probability
Complement Rule
The probability that a bag of mixed vegetables selected is underweight, P(A), plus the probability that it is not an underweight bag, written P(∼A) and read "not A," must logically equal 1. This is written: P(A) + P(∼A) = 1 This is the complement rule. It is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1. This rule is useful because sometimes it is easier to calculate the probability of an event happening by determining the probability of it not happening and subtracting the result from 1. Notice that the events A and ∼A are mutually exclusive and collectively exhaustive. Therefore, the probabilities of A and ∼A sum to 1.
Which of the following numbers could represent a probability that an event will occur? a. 5/4 b. 1.2 c. 25% d. 3/4 e. -0.3 f. .43 g. 110%
25%, 3/4, .43
an urn contains two red and three yellow balls. two balls are selected randomly. what is the probability that both are yellow?
3/10 P(y1 and y2) = P(y1)P(y2|y1) = 3/5 x 2/4
a bowl contains three red and four yellow marbles. you randomly select two marbles from the bowl. which of the following is a conditional probability? assume the second marble is drawn from the marbles remaining after the first draw
the probability that the second marble will be red, if the first one was yellow
Which one of the following is true about events that are both mutually exclusive and collectively exhaustive?
the sum of their probabilities must be 1
GENERAL RULE OF MULTIPLICATION
P(A and B)=P(A)P(B|A)
