Question

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 $335,000. If only natural gas is hit, the income will be $130,000. 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?

Answer 1

Answer on Question #45278 – Math – Statistics and Probability

Question. 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 $335,000. If only natural gas is hit, the income will be $130,000. 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?

Solution. Let event $A$ – oil hits, $B$ – gas hits. We assume that the events $A$ and $B$ are independent so $P(A) = \frac{1}{40}, P(B) = \frac{1}{20}, P(A \cap B) = P(A)P(B) = \frac{1}{800}$. Let the random variable $\xi$ be the income for the company. Find the distribution of $\xi$.

$$P(\xi = \$335,000) = P(A) = \frac{1}{40}.$$ $$P(\xi = \$130,000) = P\{\text{only gas hits}\} = P(B) - P(A \cap B) = \frac{1}{20} - \frac{1}{800} = \frac{39}{800}.$$ $$\begin{array}{l} P(\xi = 0) = P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B) = 1 - P(A) - P(B) + P(A \cap B) = 1 - \frac{1}{40} - \frac{1}{20} + \\ + \frac{1}{800} = \frac{741}{800}. \end{array}$$

So $\xi$ has the next distribution:

$$E(\xi) = \$335,000 \cdot \frac{1}{40} + \$130,000 \cdot \frac{39}{800} + 0 \cdot \frac{741}{800} = \$14,712.5$$

The expectation of income for the drilling company is equal to

$$\$14,712.5 - \$25,000 = -\$10,287.5$$

Answer. The expectation for the drilling company is equal to $-\$10,287.5$.

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