Consider the example illustrated below of selecting a job candidate
from a pool of six equally qualified applicants. This example is
presented in the form of a Venn diagram, which uses circles and their
spatial orientation to indicate relationships between sets of
information. The group of applicants on the left all have college
degrees, while the group on the right have relevant work experience.
What is the probability of selecting a candidate who has a college
degree? Recall that probability is the ratio of the number of times an
event can occur to the total possible number of events. The event being
considered is selecting a candidate who has a college degree. There are
four candidates who fit this description. The total possible number of
events equals the number of all of the candidates who can be chosen. In
this example, there are six potential candidates.
Because four of the six candidates have a college degree, the
probability of selecting a candidate with a college degree is .66. Like
all probabilities, this falls between zero and one. It would be written
in the following way.
If an event's probability is zero, the event will not occur. For
example, consider the probability of rolling a sixsided die and having
the outcome be a seven. Because no side of the die is a seven, the
number of times this event can occur is zero. The probability of this
event is written below.
If an event's probability is one, the occurrence of the event is
certain. Recall the earlier example of flipping a coin. There are only
two possible outcomes—heads or tails. It is certain that one of these
events will occur when you flip a coin; therefore, the probability of
the outcome being heads or tails is one.
Not only is the probability of a single event between zero and one,
but the sum of the probabilities must equal one. Consider again the
coin flip. The probability of the outcome of heads equals .5; the
probability of tails also equals .5. Since heads and tails are the
only possible outcomes to the flip of a coin, they are said to be collectively exhaustive events and, therefore, their probabilities must add up to one.
Mutually Exclusive Events Two events are mutually
exclusive if the probability of both events happening concurrently is
zero. When events are mutually exclusive, you can determine the
probability of either the first or the second event occurring alone by
adding their probabilities.
Consider the pool of six job candidates again.
As you can see, the set of candidates with a college degree and the
set with work experience do not intersect. They are mutually exclusive
events. This means the probability of selecting a single candidate who
has both a college degree and work experience is zero. You would
express this fact in the following way.
If the probability of two events happening concurrently is zero, the
events are mutually exclusive. When events are mutually exclusive, you
can determine the probability of the first or second event occurring
alone by adding the events' individual probabilities.
In the case of the job candidates, you can determine the probability
of selecting either a candidate with a college degree or a candidate
with work experience by adding the individual probabilities. As the
equation below shows, four of the six candidates have a college degree
and two of the six have work experience.
The probability of either one of these mutually exclusive events
occurring is one. As you learned earlier, when the probability of an
event is one, the occurrence of that event is certain. In this case,
the probability of choosing an applicant who has either a college
degree or work experience is certain.
In this example, the events are not only mutually exclusive, but
they are also collectively exhaustive; thus, their probabilties will
add up to one. The sum of the probabilities of events that are mutually
exclusive but not collectively exhaustive will not be equal to one.
Conditional Probability
A conditional probability is the likelihood of an event given the occurrence of another event.
Consider another pool of equally qualified job candidates whose
educational and employment backgrounds are illustrated in the diagram
below.
Using this new diagram, where candidates 1 and 6 have neither a
college degree nor work experience, and candidate 4 has both a degree
and experience, consider the conditional probability below. (Note that
the vertical bar in the following probability expression stands for the
word "given.")
This conditional probability asks the following question: "If a
candidate with work experience is selected, what is the probability
that that candidate will also have a college degree?" You can answer
this question by using the following ratio.
By examining the diagram, you can see that only one of the six
candidates (candidate 4) has both a college degree and work experience.
Two of the six candidates (4 and 5) have work experience. Therefore,
the conditional probability of choosing a candidate with a college
degree, given that the candidate has work experience, is .5. You would
calculate this answer mathematically as follows.
Independent Events Events are independent of one
another if the occurrence of one event does not alter the probability
of the other event. Since conditional probability examines the
probability of two events happening concurrently, you can determine
whether events are independent by comparing their conditional
probabilities to their individual probabilities.
You can determine whether the two sets of job candidates are
independent of one another by comparing their conditional probability
to their individual probability.
If either of the following statements is true, the two events are independent.
You could use either statement to determine if the events are independent. This example will focus on the first statement.
The left side of the statement is the conditional probability of
experience, given college. It asks, "If a candidate with a college
degree is selected, what is the probability that that candidate will
also have work experience?" Only one of the six candidates has
experience and a college degree. Three of the six have a college
degree. Therefore, the conditional probability on the left side of the
statement is
The right side of the statement asks, "What is the probability of
selecting a candidate with experience?" Two candidates out of six have
experience, which means the answer to the right side of the statement is
The conditional probability of experience, given college, is equal
to the individual probability of college. This means that the first
statement is true.
Because the two sides of the statement are equal, the event of
selecting a candidate with a college degree is independent of the event
of selecting a candidate with work experience. If the two sides of the
statement were not equal, the events would not be independent.
Rule of Multiplication In business, you will often want
to be able to determine the probability of more than one event or the
probability of a sequence of events. To determine these more complex
probabilities, you need to understand the rule of multiplication.
Returning to the job candidate scenario, consider the probability of selecting two candidates who have college degrees.
You would use the rule of multiplication, as shown below, to
calculate this probability. (The first and second college selections
are noted as C_{1} and C_{2}, respectively.)
The first part of the equation is simply the probability of
selecting one candidate who has a college degree. Three of the six
candidates fit this description. The second part of the equation is the
conditional probability of selecting a second candidate with a college
degree, given one has already been selected. Because a candidate with a
college degree has already been picked, only two of the remaining five
candidates have college degrees.
You can determine the probability of selecting two college
candidates sequentially by multiplying the individual probability and
the conditional probability as shown below.
The probability of selecting two college candidates sequentially is
.2. As you can see, the probability of selecting two college candidates
is less than the probability of selecting one. The likelihood of two
events occurring sequentially is never greater than the likelihood of
one of the events occurring alone.
Applying the rule of multiplication to independent events simplifies
the math a bit. Consider independent events A and B. The rule of
multiplication states that the probability of A and B is equal to the
probability of A times the probability of B given A: P(A and B) = P(A)
× P(B  A). The rule of independence states that, if A and B are
independent then P(B  A) = P(B). Substituting this expression into the
rule of multiplication gives you the rule of multiplication for
independent events: P(A and B) = P(A) × P(B).
Consider the table below, which describes a pool of 75 job
applicants. The center cells of the table list joint events. If you
wanted to identify how many applicants had both 10 years of experience
and a degree from a Big Ten school, you would consult this portion of
the table and find that six applicants have both characteristics.
The table's right column and bottom row list totals for single
events. For example, if you were interested in any candidate with 10
years of work experience, you would consult the bottom row and find
that 23 applicants meet this qualification.
The probabilities associated with selecting these candidates are
listed in the table below. The joint probabilities in the center cells
were calculated using the rule of multiplication.
The probability of selecting a candidate with a Big Ten degree is called a marginal probability,
since its value can be read directly from the righthand margin of the
table. The marginal probability of an event is equal to the sum of the
corresponding joint probabilties for that event.
A joint probability is never greater than the probability of either
of the events occuring alone. This is because a joint probability
requires two conditions to be met rather than just one.
Consider how the probability of choosing an applicant who has both
10 years of experience and a degree from a Big Ten school compares to
the probability of choosing any applicant with 10 years of experience.
The joint probability of choosing a candidate with both 10 years of
experience and a Big Ten degree is .08. The probability of choosing any
candidate with 10 years experience is .30. The joint probability is
less than the individual probability.
You may also wish to consider the probability of selecting a
candidate with either a Big Ten degree or 10 years of work experience.
This is referred to as the union of two events, and the probability is
equal to the marginal probability of each event less the joint sum of
the probability of both events.
1. In every case where a manager must make a decision, there is a
degree of probability associated with a successful outcome. Between
what two values must this probability fall?
Solution 1
2. Why must the probability fall between these two values?
Solution 2
3. What information is necessary to calculate the conditional probability of an event, P(A  B)?
Solution 3
4. Given the following figure, what is the probability of choosing
one portfolio with both a low level of risk and a high level of growth?
Solution 4
5. Referring to the above question, how does knowledge that a
portfolio is highgrowth affect the probability associated with its
being lowrisk?
Solution 5
6. Two machines, A and B, produce the same part. Each machine is
known to produce defective parts at different levels of probability.
Ten percent of the parts produced by Machine A and 20 percent of the
parts produced by Machine B are defective. Once produced, the parts are
placed in one of three boxes on the shop floor. One box is marked
"Machine A" and contains only parts produced by Machine A. The second
box, marked "Machine B," contains only parts produced by Machine B. The
third box has no marking and may contain parts produced by Machine A or
Machine B. Each box contains 100 parts. Given this information, answer
the following questions.
a. Is the random selection of a part from the box marked Machine A an independent or dependent event?
b. Based on the percentage of defective parts, you assume that there
are 10 defective parts in the box marked "Machine A." A part is
randomly selected from this box and is determined to be not defective.
This part is not returned to the box. When a second part is chosen from
the same box, is the probability of that part being defective an
independent or a dependent event?
Solution 6a
Solution 6b
7. Consider the following table of values. If a manager with at
least 10 years of experience is chosen, what is the conditional
probability that the chosen manager is from the West Coast?
Solution 7
8. Given the following Venn diagram containing eight organizations,
determine whether the probability of choosing an electronics company
and choosing a company based in Japan are independent of each other.
Solution 8
9. Given the following set of airline ticket bidding options, what
is the joint probability of a bidder winning a ticket that has fewer
than three stops, as well as a ticket that costs $200 or less?
Solution 9
10. Given that a ticket will have three stops, what is the probability that it will cost $300 dollars or more?
Solution 10
