# 1. An oil company conducts a geological study that indicates that an exploratory oil well should...

1. An oil company conducts a geological study that indicates that an exploratory oil well should have a 25% chance of striking oil.
Part (a): What is the probability that the first strike comes on the third well drilled?
Part (b): What is the probability that the third strike comes on the seventh well drilled?
Part (c): What is the mean of the number of wells that must be drilled if the oil company wants to set up three producing wells?
Part (d): What is the variance of the number of wells that must be drilled if the oil company wants to set up three producing wells?

2. An oil company conducts a geological study that indicates that an exploratory oil well should have a 30% chance of striking oil.
Part (a): What is the probability that the third strike comes on the seventh well drilled?
Part (b): What is the mean of the number of wells that must be drilled if the oil company wants to set up three producing wells?
Part (c): What is the variance of the number of wells that must be drilled if the oil company wants to set up three producing wells?

1(a)

Let X is a random variable shows the number of well drilled till first strike. Here X has geometric distribution with parameter p = 0.25. The probability that the first strike comes on the third well drilled is

$P(X=3)=(1-0.25)^{3-1}\cdot 0.25=0.140625$

(b)

Let X is a random variable shows the number of well drilled till third strike. Here X has negative binomial distribution with parameters r=3 and  p = 0.25. The probability that the third strike comes on the seventh well drilled is

$P(X=7)=\binom{7-1}{3-1}\cdot (1-0.25)^{7-3}\cdot 0.25^{3}=0.0746$

(c)

Using distribution of part b:

$\mu=\frac{r}{p}=\frac{3}{0.25}=12$

(d)

Using distribution of part b:

$\sigma^{2}=\frac{r(1-p)}{p}=\frac{3\cdot 0.75}{0.25}=9$

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