y 3. Suppose that p - P(success in a single trial of an exploratory oil drilling
ID: 3357029 • Letter: Y
Question
y 3. Suppose that p - P(success in a single trial of an exploratory oil drilling) 0.4. The drilling is stopped when two successes are reached. 5) a) What is the probability that the required number of trials is less than 5? [12] b) The cost of a single trial is $10,000. The company starts driling only if the expected total cost would not exceed $60,000, and, simultaneously, if the probability that the total cost would exceed $100,000 is less than 0.1. Calculate the expected total cost, the probability that the total cost would exceed $100,000 and determine whether the company will start drilling.Explanation / Answer
a)
This is the case of negative binomial experiment which consists of x trials and results in r successes ( r - 1 successes after trial x - 1 and r successes after trial x) . If the probability of success on an individual trial is P, then the negative binomial probability is:
P(X = x) = x-1Cr-1 * Pr * (1 - P)x - r
In the problem, number of successes , r = 2 and Proabability of success, P = 0.4
Probability that number of trials is less than 5 is,
P(X < 5) = P(2) + P(3) + P(4) {Note that the number of trials should be atlest 2 for 2 successes}
= 2-1C2-1 * 0.42 * (1 - 0.4)2 -2 + 3-1C2-1 * 0.42 * (1 - 0.4)3 -2 + 4-1C2-1 * 0.42 * (1 - 0.4)4 -2
= 0.16 + 2 * 0.16 * 0.6 + 3 * 0.16 * 0.36
= 0.5248
b)
Mean of negative binomial distribution = r /P = 2 / 0.4 = 5
So, expected number of trials for 2 successes = 5
Expected total cost = 5 * $10,000 = $50,000
Probability that the total cost would exceed $100,000 = Probability that the number of trials exceed 10
= Probability that number of unsuccessful trials exceeds 9 < (1-0.4)9 = 0.69 = 0.01
So, Probability that the total cost would exceed $100,000 < 0.01
As, both conditon satisfies, the company will start drilling.
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