Question

1. an oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil. what is the probability that the first strike comes on the third well drilled? 0.128 9. what is the probability that the third strike comes on the seventh well drilled? 0.049 10. what is the mean and variance of the number of wells that must be drilled if the oil company wants to set up three producing wells?

Explanation:

Step1: Identify the distribution

This is a negative - binomial distribution problem. The probability mass function of a negative - binomial distribution is $P(X = n)=\binom{n - 1}{k-1}p^{k}(1 - p)^{n - k}$, where $k$ is the number of successes, $n$ is the number of trials, and $p$ is the probability of success in a single trial.

Step2: Solve for the mean and variance of the negative - binomial distribution

The mean $\mu$ and variance $\sigma^{2}$ of a negative - binomial distribution with parameters $k$ (number of successes) and $p$ (probability of success in a single trial) are given by the formulas:

• Mean: $\mu=\frac{k}{p}$
• Variance: $\sigma^{2}=\frac{k(1 - p)}{p^{2}}$

Here, $k = 3$ (number of producing wells) and $p=0.2$.

• Calculate the mean:
• $\mu=\frac{3}{0.2}=15$
• Calculate the variance:
• First, find $1 - p=1 - 0.2 = 0.8$.
• Then, $\sigma^{2}=\frac{3\times0.8}{0.2^{2}}=\frac{2.4}{0.04}=60$

Answer:

Mean: $15$, Variance: $60$