Risk Assessment

Risk assessment is the stage in our risk management process where the importance of each risk is evaluated. This evaluation will also serve as the guideline for determining the risk strategy. Here we use the list of identified risks that were made as inputs. The list of risks will constantly change as well, since the time of the risk and the progress toward completion of the project will affect the risks that will be on the list of identified risks.

It is critical that the risks be evaluated, since, because of risk tolerance of the stakeholders, some risks will be ignored while others will have rather elaborate monitoring and mitigation plans associated with them. The evaluation or assessment process is necessary to itemize these risks into a ranking that will place them in the order of importance.

In the evaluation process we will be concerned with determining the impact and probability of the risk. From these two factors we can determine the severity of the risk. The severity of the risk will allow its ranking in order of importance.

Qualitative Risk Analysis

Qualitative analysis is a fast, inexpensive way to organize risks according to their importance. Since it is fast and inexpensive it also means that not a lot of detailed information is collected about the risk. This can lead to errors that may rank some risks higher or lower than they deserve to be. As we have said earlier, we must perform risk analysis many times during the course of the project. Risks that are identified long before they can happen need not have a rigorous quantitative analysis since they may never actually happen. Circumstances can often change to make risks that were menacing become trivial or nonexistent.

In analyzing a risk, its probability and impact can be determined in its simplest form by stating its probability as ''likely'' or ''not likely,'' ''bad impact'' or ''not so bad.'' We can easily raise the level of discrimination by evaluating the probability and impact of risk as ''high,'' ''medium,'' or ''low.'' This raises the choices of category for a risk from two to three. We could also assess probability by assigning a number from 1 to 10, where 1 is least probable and has least impact, and 10 is very probable and has high impact. As our probability or impact discriminator becomes better, the cost and difficulty of assigning numbers becomes higher. Finally, the most discriminating analysis would be the use of specific probability estimates between zero and one, with accuracy to as many decimal places as can be estimated. Impacts can then be evaluated in terms of dollars.

Many versions of the probability and impact matrix have been done. As can be seen in Figure 7-2, the matrix is formed by assigning a value from 1 to 9 for our assessment of the risk's probability and another value from 1—9 for the risk's impact. By multiplying the values together we get a composite number called severity. The matrix gives a value for severity that can be used to value the importance or severity of the risk. Values of 28 to 81 should be considered high risks. Values from 1 to 9 should be considered low risks, and values from 10 to 27 should be considered medium risks. Expanding the concept to the point of being ridiculous, we could have these matrixes done for each risk and for each type of problem that could develop. Thus we might have separate matrices for cost, schedule, and scope impact, and so on.

Risks can additionally be categorized by their immediacy. Risks that are imminent should have a higher priority than those that are going to happen far in the future. Many of the risks anticipated far in the future may not take place at all.

Risk Tolerance

Risks that have very high probabilities but very low impacts, as well as risks that have very high impacts but very low probabilities, are risks that may not be considered as being

Figure 7-2. Risk probability and impact matrix.

7-9

7-27

28-54

49-81

O

Low

4-6

4-18

16-36

28-54

o

Medium

1-3

1-9

4-18

7-27

o

High

1-3

4-6 Impact

7-9

important to the project. It is the combination of probability and impact that causes the risk to be an important consideration to the project.

Consider a risk of very high impact and very low probability, such as the threat to a project caused by a category five hurricane occurring when and where the project work is taking place. This is probably a risk that we would not spend much time and effort worrying about. Although the problems that would occur if the office building were to be blown down or flooded during a hurricane would be great, their likelihood is low enough that we would not worry too much about the risk. Even in New Orleans, where hurricanes are more likely to occur than many other places, they have disrupted business only three times in twenty years, and even then damage was minor.

Let's look at the other extreme of risk: a risk that has a very high probability of occurring but a very low impact. An example of this kind of risk is one person on the project team calling in sick during the project. The probability that this will happen at least one time in the course of the project is close to 100 percent, but the impact is very small. This is a risk that we are not going to spend much time worrying about.

The risks we need to worry about are all the ones that have a reasonably high probability and a reasonably high impact. But what is reasonably high? As in many of the questions that are raised in project management, the answer is: It depends. In this case it depends on risk tolerance.

Risk tolerance is the willingness or the unwillingness of a person or an organization to accept risk. Some individuals and some organizations are risk takers, while others are risk avoiders. One company might be willing to take a large risk of losing a great deal of money on the chance that an even greater amount of money could be made. Another company might not be willing to take the risk of losing the money.

I do this experiment in most of the classes I teach in risk management. I ask all of the students to stand up. I then offer to make a bet with them. I have a single die that I will roll. If the rolled die comes up with a 1 or a 2, I win; if it comes up 3, 4, 5, or 6, then the student wins. The probability of winning is 2 chances in 3, and the probability of losing is 1 chance in 3.

I ask the students, ''How many of you would be willing to bet 25 cents on this roll of the dice?'' Nearly everyone stays standing. Then I ask, ''How many of you would be willing to bet $2 on this roll of the dice?'' A few sit down.

Next I ask, ''How many of you would be willing to bet $20 on this roll of the dice?'' More sit down. The next question is, ''How many of you would be willing to bet $200 on this roll of the dice?'' Many sit down.

The bet is raised until all sit down. There is usually one in the class who just keeps standing until the bet gets so high that having to pay it off would be impossible.

The people who sit down in the early stages are the risk avoiders. The people who sit down last are the risk takers. Even though the probability of winning is 2 out of 3, as the size of the bet increases people refuse to take the bet. This is because there is still some chance of losing, and the damage to their financial security is too great to be tolerated. It is one thing to bet your lunch money and have to skip lunch if you lose, but it is quite another to bet your house and car and have to do without them if you lose.

Companies work like individuals in this area of risk tolerance. Companies like General Motors have a relatively low tolerance for accepting risk. Their reluctance to accept risk was nearly their undoing in the 1970s when they were forced to compete with Japanese automakers. Other companies are willing to take large risks and even put the entire company at great risk. Amazon.com is an example of a company that is willing to risk a great deal to make a lot of money. At this writing Amazon.com, a billion-dollar (in sales) corporation, has only recently made its first profit in its entire history. Still, investors think that even though the risks are high, there is a good chance to make a lot of money on this company.

As we move into more detail in determining the components of risks to our project, it is important to realize that the techniques discussed can be used to determine all of the components of risk. When I discuss an item like expert interviews or Crawford slip, it is important to realize that these techniques could be used to help determine the probability and the impact of the risk as well as to identify the risk.

Risk Probability

Since all risks have a probability of greater than zero and less than 100 percent, the probability of a risk occurring is essential to the assessment of the risk. Any risk event that has a probability of zero cannot occur and need not be considered as a risk. A risk event that has a probability of 100 percent is not a risk. It has a certainty of occurring and must be planned for in the project plan. Understanding the fundamentals of probability is necessary if we are to understand the role of probability in risk analysis.

Probability is the number that represents the chance that a particular outcome will occur when the conditions allow it. The probability of one possibility occurring is 1 divided by the number of other possible outcomes. In the rolling of die (half a pair of dice), there is 1 chance out of 6 that the result will be a 1. The possibilities are 1, 2, 3, 4, 5, or 6. The probability would be expressed as 1/6 or .167 or even 16.7 percent. All of these terms are commonly used in expressing probability. The outcome of an event is the result of any event taking place. Outcomes cannot be subdivided into smaller outcomes: 1, 2, 3, 4, 5, and 6 represent all of the possible outcomes of rolling our die. The term event means the set of all the possible outcomes that could occur.

These events of rolling the die are considered to be mutually exclusive. A mutually exclusive event is one where the occurrence of one of the possibilities eliminates the possibility of the others. In our rolling of the die, all of the possibilities are mutually exclusive, since the rolling of a 5 makes it impossible for a 1, 2, 3, 4, or 6 to occur.

In risk management we frequently have mutually exclusive risks. A risk will occur or it will not occur. If it occurs then it cannot not occur. The sum of the probabilities of all the things that can occur in a given set of circumstances will equal 1.0. This is the sum of all the probabilities of all the possible outcomes. In our rolling of the die, all the probabilities that exist are that the die will come up with a 1, 2, 3, 4, 5, or 6. Each of these has a probability of 1/6. The sum of all the probabilities is: 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 = 1. In decimals, this is: .167 + .167 + .167 + .167 + .167 + .167 = 1.

In the rolling of a die or in situations where there are only specific outcomes possible, it is assumed that the die is a fair die, that all of the sides weigh the same, and that the corners and anything else affecting the roll of the die are equal. If we had a real die, however, there would be some bias in the die. After rolling the die many times it might be seen that one number comes up more frequently than another. This would be an indication of bias in the die.

The relative frequency definition of probability says that the proportion of past circumstances in which this outcome has occurred determines the probability of a particular outcome. This is the type of evaluation that we use most of the time in business decisions. A business is much better off relying on past experience and observations than assuming that there is no bias in the relevant circumstances. When the relative frequency definition of probability can be used, it is extremely valuable. There are two major difficulties associated with this measure, however.

Let's take the example of a business that operates a theater. The business owner must determine whether it would be better to offer a comedy or a high drama. The risk is that if the owner chooses the wrong type of entertainment, the public will not come to the theater.

Based on past experience, the theater operator determines that when a comedy was offered, money was made 80 percent of the time. When a drama was offered, money was made 60 percent of the time. Based on this past experience, the theater owner should offer nothing but comedy. But this would be a mistake, because the audience changes with each situation. In this situation, the audience would become bored with the constant offering of comedies, and the theater operator would probably lose more money than if dramas were also presented.

The other difficulty in risk management decisions using the relative frequency definition of probability is that there is usually not enough experience in the past to reliably predict the future. For this reason, subjective probability is used. Subjective probability is the best estimate that can be made to predict the future outcome based on the conditions that can be known.

As in all mathematical areas there are rules that must be followed if we are going to achieve consistent results. We will need to worry about only a few of these rules.

The Addition Rule

The addition rule in probability explains the probability of one or more of several events occurring. I will start by showing the probability of two events. We can consider the probability of either event A or event B occurring when they are both mutually exclusive events.

Suppose we roll one die again. This time we are interested in two events. Event A is that the die will come up with an odd number. Event B is that the die will come up with an even number. Event A is satisfied if there is a 1, 3, or 5. Event B is satisfied if there is a 2, 4, or 6.

The probability we are interested in is the probability that either event A or event B will occur. This means that if the die comes up with a 1, 2, 3, 4, 5, or 6 we will meet the conditions of the question. The probability of this is 1.0, since there are only six possibilities, and all six are contained in event A or B. Also notice that the probability of event A is .5 (three out of six possibilities) and the probability of event B is .5 as well.

So, for mutually exclusive events we can say that the probability of either of them occurring is the sum of the probabilities of each:

With the same explanation we could extend this to more than two mutually exclusive events. In this example, event A would be to roll a 1 or 2. Event B would be a 3 or 4. Event C would be a 5 or 6. The probability of A or B or C would be the sum of the probability of A, B, and C:

In fact, we can say that this is true for any number of mutually exclusive events: P(A or B or ... or N) = P(A) + P(B) + ... + P(N)

Well, all this is fine for probabilities that are mutually exclusive, but what about the situations where the events we are interested in are not mutually exclusive? When the events are not mutually exclusive, applying the addition rule as stated will produce a probability that is too high.

Suppose we are interested in two events with our die again. Event A is the rolling of an odd number, that is, 1, 3, or 5. Event B is the rolling of a number less than 4, that is, 1, 2, or 3.

The probability of event A is 3/6. The probability of event B is 3/6. If we apply the mutually exclusive addition rule the result would be: 3/6 + 3/6 = 6/6 = 1. Since rolling 4 or 6 are possibilities, the probability of getting an odd number or a number less than 4 cannot be 1.

The problem here is that the two events described are not mutually exclusive. It is possible to have numbers show on the die that satisfy both events at the same time. The numbers 1 and 3 showing on the die are odd and less than 4. When we look at the probability of event A, 1,3, or 5, and event B, 1,2, or 3, and apply the addition rule, we get 1, 3, 5, 1, 2, or 3. We can roll only one number in one roll of the die and the numbers that will satisfy either event A or event B are 1, 2, 3, or 5. We should not count the extra 1 and 3. The probability of event A or B is 4/6, or 2/3.

Now the addition rule can include events that are not mutually exclusive:

One more example. Suppose we roll a pair of dice, and we want to know the probability of getting at least one 6.

By the addition rule we might say that the probability of getting a 6 on the first roll is 1/6, and the probability of getting a 6 on the second roll is 1/6. The probability of getting a 6 on the first roll or the second roll is 1/6 + 1/6 = 2/6. But this would be incorrect.

In Table 7-2, all of the possible combinations of rolling two dice are shown. If we count the boxes that contain at least one 6 and divide by the number of possible combinations, we should have the probability that we seek.

Doing this we find that there are thirty-six possible outcomes for rolling a pair of dice. In the table, eleven of the combinations would produce at least one 6.

Correctly applying the addition rule, we get the following:

P(at least one 6 in two rolls) = P(6 on the first roll)

+ P(6 on the second roll) - p(6 on both rolls) P(6 or 6) = 1/6 + 1/6 - 1/36 P = 11/36

The Multiplication Rule

So far we have not explained the term P(A and B) completely. Before we explain this rule of probability, a few definitions are in order: conditional probability and statistical independence.

Conditional probability is the probability of an event given the information that some other event has occurred. Suppose we are interested in the probability of rolling a single die and getting a number less than 4. The probability is 1/2.

Now suppose that I knew that the die had an odd number on it. Does this change the probability of having a number less than 4? There are two numbers that are less than 4 and odd—1 and 3—and there are three numbers that are odd: 1, 3, and 5. If we know that the number facing up on the die is odd, we can say that the probability of it being less than 4 is 2/3, not 1/2. The condition of ''being odd'' has changed the probability. This is conditional probability. The notation that is used, P(A | B), is read, ''The probability of event A given event B.''

In our example, event A was a number less than 4, and event B was an odd number. Our statement was the probability of having a number less than 4 given that the number is an odd number.

Statistical independence says that the probability of event A is the same as the probability of event A given the probability of event B. If P(A) = P(A | B), then event A and event B are statistically independent.

The probability of rolling a 6 on the third roll of a die is 1/6. If a 6 had been rolled on the second or even the first and second roll of the die, the probability of getting a 6 on the third roll is still 1/6. These events, rolling a die on the first roll, rolling a die on the second roll, and rolling a die on the third roll are statistically independent.

Table 7-2. Possible combinations of rolling two dice.

1,1

1,2

1,3

1,4

1,5

1,6

2,1

2,2

2,3

2,4

2,5

2,6

3,1

3,2

3,3

3,4

3,5

3,6

4,1

4,2

4,3

4,4

4,5

4,6

5,1

5,2

5,3

5,4

5,5

5,6

6,1

6,2

6,3

6,4

6,5

6,6

Statistically dependent events are events where the probability of A given B is not the same as the probability of A alone. The probability of getting a number less than 4 given that the number is odd is different from the probability of getting a number less than 4. In this case it is 2/3 versus 1/2. Therefore, these events are statistically dependent.

Finally, getting back to the multiplication rule:

The probability of getting event A and event B is the probability of event A given event B multiplied by the probability of event B.

In the example of the two dice being rolled and finding the probability of getting at least one 6, we had to subtract the probability of getting a 6 and a 6 on a roll of two dice. This value can be calculated by seeing that the P(6 on the second die), 1/6, and the P(6 on the first die given that a 6 was rolled on the second die), 1/6:

P(A and B) = P(A | B) P(B) P(A and B) = 1/6 X 1/6 = 1/36

Notice that event A and event B are also statistically independent. This means that we could have stated the equation this way:

This simplified form of the multiplication rule for statistically independent events is useful. Suppose we want to know the probability of rolling a 6 on a die three times in a row. Since the events are statistically independent, the multiplication rule can be applied:

P(6 on the first roll and 6 on the second roll and 6 on the third roll)

For example, suppose we are interested in the risk of not having a critical part delivered for our project. The vendor that we have chosen has a reliability of delivering parts on time 95 percent of the time. This is an unacceptable risk for the project, and the project manager decides that the order will be split between two different vendors. The project manager hopes that at least one of the vendors will deliver on time.

The probability that vendor A will be late is .05. The probability that vendor B will be late is .05. Events A and B are statistically independent, since the delivery of one vendor on time does not influence the probability of the other vendor delivering on time. The probability of both vendors being late is:

P(A being late and B being late) = P(A being late) X P(B being late)

P(A and B) = .05 X .05 = .0025, or 1/4 of 1 percent

To summarize, the assessment of the probability of the risk is important for determining the overall importance of the risk. The rules of probability are important to know so that there is an understanding about what these values mean and that we have some guidance in determining the values to use. Most project risks can be considered mutually exclusive, but not always. Mutually exclusive risks are those where we consider the probability event as ''the risk will occur or it will not.'' If the risk occurs, then it cannot not occur. This meets our definition of mutually exclusive.

Risk Impact

The next subject we need to discuss is the next component in risk management, the evaluation of the impact of the risk. Risk impact is the cost of the risk if it occurs. This, in its qualitative measure, is the pain level of the risk. Quantitative measures include the impact of the identified risk in terms of schedule days, effort person-hours, money, and so on.

Risk impacts are those things that affect the cost, schedule, or scope of the project. These impacts can manifest themselves as effects on the level of effort required, labor rates, duration of tasks, technical feasibility, material suitability, material cost, equipment availability, and more.

In determining the impact of a risk it is important to realize that all of the techniques that we have previously discussed, such as brainstorming or the use of probability analysis, can be used to determine the impact of a risk. Likewise, the use of the tools discussed here is not limited to their use in impact analysis. They may yield valuable information about risks that have not been previously identified.

In its simplest terms, impact can be described as ''real bad'' or ''not so bad.'' This separates risks into those that we think have a great impact and those that we think do not. We could improve this by addressing impacts as ''high,'' ''medium,'' or ''low.'' We could further improve this evaluation by giving the risk a numerical value from 1 to 10, or even 1 to 100.

Expected Value

Expected values are a way of combining the probability and the impact of a risk in a meaningful way. The expected value calculation is simply multiplying the probability, in terms of zero to one, times the impact, usually measured in terms of dollars or schedule days. Impact may be measured in any convenient value.

Since this is a more quantitative result than the usual subjective values of probability and impact, it is proper to summarize expected values to total project risk. For example, consider the possibilities of winning money on a lottery ticket. The ticket you buy can win $2, with a probability of 5 percent. It could also win $100 with a probability of 1/2 percent. Of course, there is a 94.5 percent chance of winning nothing. The ticket costs $1.00 to play.

Notice that the three possible outcomes of the event are mutually exclusive. If you were to win $2, it would not be possible to win $100 or nothing (Table 7-3).

With expected values we have a way of evaluating the opportunities and risks involved in the project. The expected value is also a good guideline for the amount of money that might be spent to eliminate the risk.

Let's say that there is a 10 percent chance of a risk occurring that would have a $10,000 impact on the project. The expected value of this risk would be $1,000. If it would be possible to completely avoid this risk by spending $900, it would be considered a good decision to avoid this risk.

Another way of looking at the expected value is to think of the project as being done many hundreds of times (theoretically, that is). For a risk that has a probability of 10 percent, the risk would probably occur in 10 percent of the projects. The average cost of the risk to all the hundreds of projects would be 10 percent of the total risk impact.

It is also interesting to look at the best-case and worst-case situations for the project. This is a simple analytical method that gives us insight into the extreme possibilities that might occur in the project. This is useful when considering the risk tolerance of the individuals or groups involved in the decision-making process.

In the best-case expected value calculation, all of the positive risks are considered to have happened, while none of the negative risks are considered. In the worst-case expected value calculation, all of the negative risks are considered to have occurred, while none of the good risks are considered.

For example, suppose after analyzing risks of a potential project we find the situation shown in Table 7-4. Notice that in the calculation of the worst-case and best-case situations the probability of the risks is not considered. The best case is where everything good that can happen happens and everything bad that can happen does not happen. The worst case is where everything bad that can happen happens and everything good that can happen does not happen.

Table 7-3. The possibilities of winning money on a lottery ticket.

Probability

Impact

Expected Value

.05

2

.10

.005

100

.50

.945

0

0

Total expected value of

revenue

.65

Cost of ticket

-1.00

Expected value of the

opportunity

-.35

Table 7-4. Worst-case

and best-case situations.

Risk Event

Impact

Probability

Expected Value

Project cost

-2,000,000

-2,000,000

Project revenue

2,200,000

+ 2,200,000

Fail acceptance test

-100,000

10%

-10,000

Warranty failures

-40,000

15%

- 6,000

Additional orders

75,000

30%

+ 22,500

Penalty for late delivery

-50,000

5%

-2,500

Incentive for early

100,000

30%

delivery Expected value of the project (sum of all values) Best case (all good risks occur, no bad risks occur) Worst case (all bad risks occur, no good risks occur)

234,000

375,000

10,000

Decision Trees

In a more complex situation it is difficult to calculate the expected value of the project. For these more complex situations a technique called decision tree analysis is often used. In this case a large number of individual outcomes are possible.

For example, let's say that you have a large uncut diamond of 6 carats. The diamond cutter says that if the diamond is cut into small stones, the aggregate value of the stones will be $250,000. If the diamond is cut into one large stone, the value will be $100,000. The problem associated with cutting the diamond into smaller stones is that there is a 20 percent chance that the diamond will shatter when cut. If the diamond shatters when it is cut, the aggregate value will be $10,000.

In making the decision to cut or not cut the diamond, expected values could be used. There is a 20 percent chance that the diamond will be worth $10,000, and there is an 80 percent chance that it will be worth $250,000. The expected value of these two mutually exclusive possibilities is:

If the diamond is not cut, the expected value is $100,000. The obvious choice is to have the diamond cut into smaller diamonds. The decision tree diagram for this situation is shown in Figure 7-3. In the decision tree diagram, boxes are used to represent decisions that can be made, and circles are used to indicate probabilistic events that may occur.

Figure 7-3. Decision tree.

$250(000

$100(000

$10(000

Decision:

To cut or not to cut

$250(000

$100(000

Suppose we now complicate the process. For a $5,000 fee, the diamond can be sent to a firm that can study the structure of the diamond with an electron microscope and microsound echo scanning to improve the chances of cutting the diamond successfully. According to the firm proposing the study, if they predict that the diamond will not shatter, then 99 percent of the time, when the diamond is cut, it will not shatter. If the prediction is that the diamond will shatter, then the diamond will shatter 95 percent of the time. Let's say that the diamond itself has a 20 percent chance of shattering, as before.

The decisions to be made are: Should you pay for the prediction, and should you have the diamond cut?

The decision that must be made is whether to pay for the inspection. Regardless of whether the inspection is performed, a decision must still be made as to whether to have the diamond cut or not. If the decision is made not to go ahead with the inspection, then the choices are the same, with the same expected values that we had in the simpler example. Once the inspection is completed, it will predict 20 percent of the time that the stone will shatter, and it will predict that 80 percent of the time the stone will not. This is not smoke and mirrors; of all the diamonds cut in recent times, 20 percent of this type of diamond have shattered. The question is whether this particular diamond will shatter. That is the point of the inspection.

In the upper part of Figure 7-4, the decision has been made to purchase the inspection. Twenty percent of the time the inspection will predict shattering, and 80 percent of the time the inspection will predict not shattering. Of course, if the inspection predicts shattering, there is a 5 percent chance that the diamond will not shatter when cut anyway. If the inspection predicts that the diamond will not shatter, there is a 1 percent chance that it is wrong and the diamond will shatter anyway.

Figure 7-4. Cutting the diamond.

Figure 7-4. Cutting the diamond.

Dont cut

$100,000

So, what decision should be made?

In the choice to not cut the stone after the lab predicts that it would shatter, the expected value of the decision is $95,000. This is because deciding to cut the diamond under these conditions yields an expected value of $14,600: the $10,000 value of the shattered stone minus the $5,000 fee to the inspection company, or $250,000 less the $5,000 fee. Ninety-five percent of $5,000 plus 5 percent of $245,000 equals $17,000.

The decision not to cut the stone yields $95,000, the $100,000 value of the uncut stone minus the $5,000 fee for the inspection. The decision is made to not have the stone cut after the inspection predicted shattering of the stone.

If the inspecting company predicts that the stone will not shatter, you still must make the decision whether or not to cut the stone. If the decision is made to not cut the stone, the yield is $95,000, the $100,000 value of the uncut stone minus the $5,000 fee for the inspection.

If the decision is made to cut the stone, the expected value is $242,600. If the stone is not cut, the value is $95,000. If the stone is cut, there is a 1 percent chance that it will shatter, yielding $5,000. There is a 99 percent chance that the stone will not shatter, yielding $245,000. The expected value of cutting the stone is:

The decision to cut the stone yields $242,600.

Moving to the next branching in Figure 7-4, there is a 20 percent chance that the inspection will predict shattering and an 80 percent chance that it will predict not shattering.

The expected value is:

The last decision is whether to hire the inspection or not. Since the expected value of not having the inspection yielded a value of $202,000, and the expected value of the decision to have the inspection done is $213,080, the inspecting company should be hired.

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Project Management Made Easy

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