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Chapter 16: Problem 30

Do the problems using the expected value concepts. A Texas oil drilling company has determined that it costs $$\$ 25,000$$ to sink a test well. If oil is hit, the revenue for the company will be $$\$ 500,000$$. If natural gas is found, the revenue will be $$\$ 150,000$$. If the probability of hitting oil is \(3 \%\) and of hitting gas is \(6 \%,\) find the expected value of sinking a test well.

Short Answer

Expert verified
The company is expected to lose \$1000 every time it sinks a test well.

Step by step solution

01

Identify and Define Variables

Define variables for the cost of sinking a well \(C = \$ 25,000\), the revenue for finding oil \(R_{o} = \$ 500,000\), the revenue for finding gas \(R_{g} = \$ 150,000\), the probability of finding oil \(P_{o} = 0.03\) and the probability of finding gas \(P_{g} = 0.06\).
02

Calculate the Expected Values

Compute the expected value of the revenues by multiplying the revenue of each outcome by their probabilities and adding them together. \(E_{o} = P_{o} \cdot R_{o} = 0.03 \cdot \$ 500,000 = \$ 15,000\) and \(E_{g} = P_{g} \cdot R_{g} = 0.06 \cdot \$ 150,000 = \$ 9,000\).
03

Summarize the Expected Value

Sum the expected revenues and subtract the cost of sinking a well. \(E = E_{o}+E_{g}-C = \$ 15,000 + \$ 9,000 - \$ 25,000 = -\$ 1,000\)
04

Interpret the Result

Since the expected value is negative, the company is predicted to lose \$1000 on average each time it sinks a test well, taking into account the probabilities of hitting oil or gas.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculations
Understanding probability calculations is essential for analyzing potential outcomes in uncertain situations. Probability is a measure of the likelihood that a particular event will occur. In the exercise, we have two key probabilities: finding oil and finding gas.
  • Finding Oil: The probability is given as 3%, which can be converted to a decimal as 0.03.
  • Finding Gas: The probability is 6%, equating to 0.06 in decimal form.
To calculate the expected value later, these probabilities are used to weigh the potential revenues from each scenario. Remember that probability values range from 0 to 1, where 0 indicates impossibility, and 1 means certainty.
They鈥檙e crucial in determining how often certain outcomes happen under repeated trials.
Revenue Prediction
Revenue prediction in this context involves estimating the potential earnings from drilling a test well. This estimation helps the company make informed decisions.
  • If Oil is Found: The revenue is $500,000, which represents a significant gain given the expense of drilling.
  • If Gas is Found: The revenue is $150,000, which still offers a profit but less compared to finding oil.
  • If Neither is Found: No additional revenue is generated from this scenario.
These predicted figures are then used alongside probabilities to determine an expected prediction through the process of finding an expected value.
This helps in understanding whether the operation is financially viable over long periods.
Risk Assessment
Risk assessment involves understanding the potential losses and gains and the likelihood of these occurring. By calculating expected values, the company assesses whether it鈥檚 worth investing in drilling a new well.
  • The likelihood of not hitting either oil or gas (91%) plays a significant role in risk.
  • The negative expected value calculated (-$1,000) indicates that on average they will lose money.
  • This gives a clear picture of the most probable outcomes and their financial implications.
Such assessments are vital in devising strategies that mitigate losses while maximizing potential gains under uncertainty.
Cost Analysis
Cost analysis involves understanding the expenses involved in a business venture. Here, the primary cost is the $25,000 expense for sinking a test well.
  • This figure needs to be offset by potential revenues to determine profitability.
  • In this exercise, the costs outweigh potential expected revenues, leading to a negative result.
  • Cost analysis helps in understanding at what point the project becomes profitable, if at all.
It鈥檚 a critical component for any business as it identifies whether revenue predictions are sufficient to cover these costs, ensuring sustainable financial health over time.

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