Question

A geological study indicates that an exploratory oil well drilled in a certain region should strike oil with probability 0.1. Assume that striking oil (viewed as success) at one drilling location is independent of success at another location.
(a) Let X be the number of successful strikes if 8 wells are dug. What is the distribution of X.

(b) What is the probability of obtaining at least (inclusive) 3 oil strikes if 8 wells are dug?

(c) If 40 wells are drilled, what is the approximate probability that 8 or fewer will strike oil? Justify your choice of the approximating distribution by verifying the relevant condition(s).

Answer 1

Answer:

(a) The distribution of X is Binomial distribution.

(b) The probability of obtaining at least 3 oil strikes if 8 wells are dug is 0.0381.

(c) The probability that 8 or fewer wells will strike oil is 0.9876.

Step-by-step explanation:

Let random variable X = number of exploratory oil well drilled in a certain region should strike oil.

(a)

The probability of successful strike is, P (X) = p = 0.10.

The number of wells dug is, n = 8.

The outcome of the random variable X are:

1. Successful strike.
2. Unsuccessful strike.

The event of striking oil at one drilling location is independent of success at another location.

The random variables satisfies all the properties of a binomial random variable.

Thus, the distribution of X is Binomial distribution.

(b)

The probability function of a Binomial distribution is:

$P(X=x) ={n\choose x}p^{x}(1-p)^{n-x};\ x=01,2,3,...$

Compute the probability of obtaining at least 3 oil strikes if 8 wells are dug as follows:

P (X ≥ 3) = 1 - P (X < 3)

              = 1 - P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)

              $=1-{8\choose 0}(0.10)^{0}(1-0.10)^{8-0}-{8\choose 1}(0.10)^{1}(1-0.10)^{8-1}\\-{8\choose 2}(0.10)^{2}(1-0.10)^{8-2}\\=1-0.4305-0.3826-0.1488\\=0.0381$

Thus, the probability of obtaining at least 3 oil strikes if 8 wells are dug is 0.0381.

(c)

A Poisson distribution is used to approximate the Binomial distribution  when the following conditions are satisfied:

$np\leq 10\\n\geq 20\\P(Success)\ is\ small$

The sample size is, n = 40.

The P (Success) = p = 0.10 (small)

Check the conditions as follows:

$np=40\times0.10=4$

n = 40 > 20.

Thus, a Poisson distribution can be used to approximate the Binomial distribution of the random variable X.

The random variable X thus follows a Poisson distribution with parameter $\lambda=np=4$.

The probability function of a Poisson distribution is:

$P(X=x)=\frac{e^{-\lamda}\lambda^{x}}{x!};\ x=0,1,2,...$

Compute the probability that 8 or fewer wells will strike oil as follows:

P (X ≤ 8) = P (X = 0) + P (X = 1) + P (X = 2) + ... + P (X = 8)

              $=\frac{e^{-4}(-4)^{0}}{0!}+\frac{e^{-4}(-4)^{1}}{1!}+\frac{e^{-4}(-4)^{2}}{2!}+...+\frac{e^{-4}(-4)^{8}}{8!}\\=0.0183+0.0733+0.1465+...+0.0298\\=0.9786$

Thus, the probability that 8 or fewer wells will strike oil is 0.9876.