An oil-drilling company knows that it costs $25,000 to sink a test well. If oil is hit, the income for the drilling company will be $395,000. If only natural gas is hit, the income will be $135,000. If nothing is hit, there will be no income. If the probability of hitting oil is 1/40 and if the probability of hitting gas is 1/20, what is the expectation for the drilling company?

Calculate expected gain, E(X), of the venture. If E(X) is positive, it would probably be advantageous to sink a well.

Expected value is the following sum:
E(x)=Σ G(X)*P(X)
where X is a possible outcome,
G(X) is the net gain of the outcome, and
P(X) is the probability of the outcome happening.
The above sum is to be summed over all possible outcomes, X.

Now make a table of all possible outcomes:
outcome X1: hit oil
revenue, G(X1): 395000-25000
probability, P(X1): 1/40
calculate E(X1)=G(X1)*P(X1)

outcome X2: hit gas
revenue, G(X2): 135000-25000
probability, P(X2): 1/20
calculate E(X2)=G(X2)*P(X2)

outcome X3: hit gas
revenue, G(X3): 0-25000
probability, P(X3): 1-1/40-1/20=37/40
calculate E(X3)=G(X3)*P(X3)

The expectation for sinking one test well is therefore
E(x)=E(X1)+E(X2)+E(X3)

* outcome 3 is "hit neither"

To find the expectation for the drilling company, we need to calculate the expected value for each outcome (hitting oil, hitting gas, and hitting nothing) and then sum them up.

The expected value is calculated by multiplying each outcome by its corresponding probability and summing up the results.

Let's start by calculating the expected value for hitting oil:

Expected value for hitting oil = (Probability of hitting oil) * (Income from hitting oil) = (1/40) * $395,000 = $9,875

Next, let's calculate the expected value for hitting gas:

Expected value for hitting gas = (Probability of hitting gas) * (Income from hitting gas) = (1/20) * $135,000 = $6,750

Finally, let's calculate the expected value for hitting nothing:

Expected value for hitting nothing = (Probability of hitting nothing) * (Income from hitting nothing) = (1 - (Probability of hitting oil) - (Probability of hitting gas)) * $0 = (1 - 1/40 - 1/20) * $0 = (37/40) * $0 = $0

Now we can calculate the overall expectation for the drilling company by summing up the expected values for each outcome:

Expectation = Expected value for hitting oil + Expected value for hitting gas + Expected value for hitting nothing = $9,875 + $6,750 + $0 = $16,625

Therefore, the expectation for the drilling company is $16,625.