The expected value is defined as the difference between expected profits and expected costs. Expected profit is the probability of receiving a certain profit times the profit, and the expected cost is the probability that a certain cost will be incurred times the cost.
Example 62:
A wheel of fortune in a gambling casino has 54 different slots in which the wheel pointer can stop. Four of the 54 slots contain the number 9. For a 1 dollar bet on hitting a 9, if he or she succeeds, the gambler wins 10 dollars plus the return of the 1 dollar bet. What is the expected value of this gambling game? What is the meaning of the expected value result?
$$\begin{array}{c}\text{ProbabilityofSuccess}=4/54\\ \text{ProbabilityofFailure}=50/54\\ \text{ExpectedValue}=\text{ExpectedProfit}\text{ExpectedCost}=\left(4/54\right)*10\left(50/54\right)*1=\$0.185\end{array}$$ 0.185 dollars indicates that if the gambler plays this game over and over again, the average gain for the gambler per bet equals  0.185 dollars, which means the gambler will lose 0.185 dollars per bet. Note that for a satisfactory investment, a positive expected value is a necessary, but not sufficient, condition.
Example 63:
Assume drilling a well costs 400,000 dollars. There are three probable outcomes:
a) 70% probability that the drilled well is a dry hole
b) 25% probability that the drilled well is a producer well with such rate that can be sold immediately at 2,500,000 dollars
c) 5% probability that the drilled well is a producer well with such rate that can be sold immediately at 4,000,000 dollars
Calculate the project's expected value.
Note that +425,000 dollars is a statistical term; it means the average of +425,000 dollars will be achieved in the longterm for drilling over and over again in a repeated investment of this type.
Expected NPV and Expected ROR Analysis
Example 64:
Assume a research project that has the initial investment cost of 100,000 dollars. There are two possible outcomes:
a) 30 % success: that leads to an annual profit of 60,000 dollars for five years (starting from year 1) with a salvage value of zero
b) 70 % failure: that leads to annual profit and salvage value of zero
Considering a minimum 12% discount rate, compare the expected NPV, and explain if this investment is satisfactory.
30 % success:  $100,000  $60,000  $60,000  $60,000  $60,000  $60,000 
70 % failure:  $100,000  0  0  0  0  


0  1  2  3  4  5 
Since considering risk in calculations results in negative expected Net Present Value (ENPV), it can be concluded that this investment is expected to be economically unsatisfactory. Note that riskfree NPV (assuming 100% success probability) shows good and economically satisfactory results.
$$\text{RiskfreeNPV:}60,000\left(P/{A}_{12\%,5}\right)100,000=\$116,287$$Example 65:
Calculate the expected Rate of Return for the above example.
Expected ROR is the “i” that makes Expected NPV equal 0.
Expected Present worth income @ "i" – Present Worth Cost @"i" = 0
$\begin{array}{c}0.3\left(60,000\left(P/{A}_{i,5}\right)\right)=0.3*100,000+0.7*100,000\\ 0.3\left(60,000\left(P/{A}_{i,5}\right)\right)=100,000\end{array}$
By trial and error, Expected ROR =  3.4%
Note that risk free ROR shows a satisfactory result.
$60,000\left(P/{A}_{i,5}\right)=100,000$
Riskfree ROR = 52.8%, which is much higher than the minimum ROR.
Another way to calculate the expected ROR, which is similar to the previous method, is to calculate expected cash flow and then find the ROR for that.
Expected cash flow can be determined by multiplying each scenario’s cash flow by its probability and then make summation over each year:
Year 1  Year 2  Year 3  Year 4  Year 5  

Expected cash flow  $$\begin{array}{c}0.3*\left(\$100,000\right)\\ +0.7*\left(\$100,000\right)\end{array}$$  $$\begin{array}{c}0.3*\left(\$60,000\right)\\ +0.7*\left(0\right)\end{array}$$  $$\begin{array}{c}0.3*\left(\$60,000\right)\\ +0.7*\left(0\right)\end{array}$$  $$\begin{array}{c}0.3*\left(\$60,000\right)\\ +0.7*\left(0\right)\end{array}$$  $$\begin{array}{c}0.3*\left(\$60,000\right)\\ +0.7*\left(0\right)\end{array}$$ 
Then:
Year 1  Year 2  Year 3  Year 4  Year 5  

Expected cash flow  $100,000  $18,000  $18,000  $18,000  $18,000 
By trial and error, Expected ROR =  3.4%
Please watch the following video (14:01): Expected Value Analysis, Part I.
Example 66:
Calculate Expected NPV for a minimum ROR 20% to evaluate the economic potential of buying and drilling an oil lease with the following estimated cost, revenues, and success probabilities.
The lease would cost 100,000 dollars at time zero and it is considered 100% certain that a well would be drilled to the point of completion one year later for a cost of 500,000 dollars. There is a 60% probability that well logs look good enough to complete the well at year 1 for a 400,000 dollar competition cost. If the well logs are unsatisfactory, an abandonment cost of 40,000 dollars will be incurred at year 1. If the well is completed, it is estimated there will be a 50% probability of generating production that will give 450,000 dollars per year net income for years 2 through 10 and a 35% probability of generating 300,000 dollars per year net income for years 2 through 10, with a 15% probability of the well completion being unsuccessful, due to water or unforeseen completion difficulties, giving a year 2 salvage value of 250,000 dollars for producing equipment.
The above decisionmaking process can be displayed in the following figure. These types of graphs are called decision trees and are very useful for risk involved decisions. Each circle indicates a chance or probability node, which is the point at which situations deviate from one another. (Costs are shown in thousands of dollars.)
Note: Times 1 and +1 are the same points in time and both indicate the end of year 1. The main body (of the tree) starts from the first node on the left with a time zero lease cost of 100,000 dollars that is common between all four situations. The next node, moving to the right, is the node that includes a common drilling cost of 500,000 dollars. At this node, an unsatisfactory and abandonment situation with a cost of 40,000 dollars in the first year (situation D) deviates from other situations (a branch for situation D deviates from the tree main body). The next node on the right (third node) is the node where situations A, B, and C (three separate branches) get separated from each other. At the beginning of each branch is the probability of that situation, and at the end of it, amounts due to that situation (including cost, income, and salvage value) are displayed.
So, there are four stations:
Situation A: Successful development that yields the income of 450 dollars per year
Situation B: Successful development that yields the income of 300 dollars per year
Situation C: Failure that yields a salvage value of 250 dollars at the end of year two
Situation D: Failure that yields abandonment cost of 40 dollars at the end of year one
Probability of situation A can be calculated as $P=0.5*0.6=0.3$
Probability of situation B can be calculated as $P=0.35*0.6=0.21$
Probability of situation C can be calculated as $P=0.15*0.6=0.09$
Probability of situation D can be calculated as $P=0.4$
Probability  
A) 
0.3

C=$100  C=$500+$400  I=$450  I=$450  ...  I=$450 
B) 
0.21

C=$100  C=$500+$400  I=$300  I=$300  ...  I=$300 
C) 
0.09

C=$100  C=$500+$400  Salvage=$250  0  ...  0 
D)  0.4  C=$100  C=$500+$40  0  0  ...  0 


0  1  2  3  ...  10 
Note that the summation of all properties should equal 1.
Project ENPV is the summation of ENPV for all situations. So, first, we need to calculate ENPV for each situation:
$$\begin{array}{c}A:\text{}\left(0.3\right)\left[100900\left(P/{F}_{20\%,1}\right)+450\left(P/{A}_{20\%,9}\right)\left(P/{F}_{20\%,1}\right)\right]\\ B:\text{}\left(0.21\right)\left[100900\left(P/{F}_{20\%,1}\right)+300\left(P/{A}_{20\%,9}\right)\left(P/{F}_{20\%,1}\right)\right]\\ C:\text{}\left(0.09\right)\left[100900\left(P/{F}_{20\%,1}\right)+250\left(P/{F}_{20\%,2}\right)\right]\\ D:\text{}\left(0.4\right)\left[100540\left(P/{F}_{20\%,1}\right)\right]\end{array}$$And it can be summarized in Table 61 as:
Probability  Year 1  Year 2  Year 3  Year 4  ...  Year 5  ENPV  

A  0.3  $100  $900  $450  $450  ...  $450  $198.5 
B  0.21  $100  $900  $300  $300  ...  $300  $33.1 
C  0.09  $100  $900  $250  0  ...  0  $60.9 
D  0.4  $100  $540  0  0  ...  0  $220 
Project ENPV is slightly less than zero compared to the total project cost of 1 million dollars, therefore, slightly unsatisfactory or breakeven economics are indicated.
Please watch the following video (13:32): Expected Value Analysis, Part 2.
Italicized sections are from Stermole, F.J., Stermole, J.M. (2014) Economic Evaluation and Investment Decision Methods, 14 edition. Lakewood, Colorado: Investment Evaluations Co.