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 Discrete Probability Distributions: Probability

In the business world, uncertainties abound. Will a particular candidate be chosen for a job? Will a training course lead to increased efficiency in a work unit? Will a company's stock price rise or fall? Business generally operates more efficiently when uncertainties are quantified through probability. If particular events have occurred in the past, information regarding their past frequency of occurrence can provide insight into their future probability of occurrence.

The probability associated with any given event is a measure of the likelihood that the event will occur. This can be calculated as a ratio of the number of times an event can occur to the total possible number of events.

To understand this ratio, imagine flipping a coin. What is the probability that the outcome will be heads? You might already know that the answer is .5. Using the ratio above, you can see why this is so. The first part of the ratio is the number of times an event can occur. In this case, the number of times the event heads can occur is one because there is only one head on the coin. The second part of the ratio is the total possible number of events. When you flip a coin, there are two possible events, heads or tails. Therefore, the probability—before the flip—of the outcome being heads would be described by the following ratio.

There are five fundamental rules of probability that will help you quantify the probability of a discrete event. Be sure to view the animated examples of each rule by selecting the "Animation" links that follow.

Probability Falls Between Zero and One

Every measure of probability falls along a continuum ranging from zero to one.

 Probability: Always Between Zero and OneView animation

Why would an entrepeneur use probability?

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 six-sided 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.

 Mutually Exclusive EventsView animation

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.

 Conditional ProbabilityView animation

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.

 Independent EventsView animation

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.

 Rule of MultiplicationView animation

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 C1 and C2, 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 right-hand 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?

2. Why must the probability fall between these two values?

3. What information is necessary to calculate the conditional probability of an event, P(A | B)?

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?

5. Referring to the above question, how does knowledge that a portfolio is high-growth affect the probability associated with its being low-risk?

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?

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?

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.

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?

10. Given that a ticket will have three stops, what is the probability that it will cost \$300 dollars or more?

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