Question

An oil well company is bidding for the rights to drill a well in field A and a well in field B. The probability it will drill a well in field A is 40%. If it does drill in field A, the probability that the well will be successful is 45%. The probability that it will drill a well in B is 30%. If it does drill in field B, the probability that the well will be successful is 55%. Drilling in field A is independent of drilling in field B and the success of the wells are independent from each other. If a well in A is successful, the company will earn $2,200,000 in gross revenue but it costs $51,000 to drill in field A. A successful well in field B will generate $2,400,000 in gross revenue for the company but it costs $68,000 to drill in field B.

a) What is the probability that the company drills a well in both fields?

b) What is the probability of the company having at least one successful oil well?

c) What is the probability that the well is located in field A given that a well is successful?

d) What is the oil company's expected profit?

Answer 1

a)

Probability that the company drills a well in both fields = probability that the company drills a well in field A * probability that the company drills a well in field B = 0.4 * 0.3 = 0.12

b)

probability of the company having at least one successful oil well =   

probability that the company drills a well in field A * probability that the company does not drills a well in field B * probability that the well will be successful in A +

probability that the company does not drills a well in field A * probability that the company drills a well in field B * probability that the well will be successful in B +

probability that the company drills a well in field A * probability that the company drills a well in field B * probability that the well will be successful in A * probability that the well will not be successful in B +

probability that the company drills a well in field A * probability that the company drills a well in field B * probability that the well will not be successful in A * probability that the well will be successful in B +

probability that the company drills a well in field A * probability that the company drills a well in field B * probability that the well will be successful in A * probability that the well will be successful in B

= 0.4 * (1 - 0.3) * 0.45 + (1 - 0.4) * 0.3 * 0.55 + 0.4 * 0.3 * 0.45 * (1 - 0.55) + 0.4 * 0.3 * (1-0.45) * 0.55 + 0.4 * 0.3 * 0.45 * 0.55

= 0.3153

c)

Let A be the event that company will drill a well in field A

Probability that the well is located in field A given that a well is successful = P(A | Successful)

= P(Successful | A) P(A) / P(Successful) (By Bayes theorem)

= 0.45 * 0.4 / 0.3153

= 0.5708849

d)

Profit when well will be successful in field A = $2,200,000 - $51,000 = 2149000

Profit when well will be successful in field B = $2,400,000 - $68,000 = 2332000

Profit when well will not be successful in field A = - $51,000

Profit when well will not be successful in field B = - $68,000  

Probability that well will be successful in field A = probability that the company drills a well in field A * probability that well will be successful in field A = 0.4 * 0.45 = 0.18

Probability that well will be successful in field B = probability that the company drills a well in field B * probability that well will be successful in field B = 0.3 * 0.55 = 0.165

Probability that well will not be successful in field A = probability that the company drills a well in field A * probability that well will not be successful in field A = 0.4 * (1 - 0.45) = 0.22

Probability that well will not be successful in field B = probability that the company drills a well in field B * probability that well will not be successful in field B = 0.3 * (1 - 0.55) = 0.135

Oil company's expected profit = Probability that well will be successful in field A * Profit when well will be successful in field A + Probability that well will be successful in field B * Profit when well will be successful in field B + Probability that well will not be successful in field A * Profit when well will not be successful in field A + Probability that well will not be successful in field B * Profit when well will not be successful in field B

= 0.18 * 2149000 + 0.165 * 2332000 + 0.22 * -51000 + 0.135 * -68000

= $7,51,200