Answer to Question #221195 in Statistics and Probability for M ishfaq

Question

Q. An oil company conducts a geological study that indicates that an exploratory oil-well should have a 30% chance of striking oil.

i. What is the probability that the first strike comes on the third well drilled?

ii. What is the probability that the driller drills at 10 locations and has 3 successes?

iii. What is the probability that the third strike comes on the eighth well drilled?

iv. What is the probability that the driller drills at 100 locations and has more than 8 successes?

Answer 1

Solution:

a) This is a geometric distribution with p=0.3.

$P(X=3)=(1-p)^2p=0.7^2\times0.3=0.147$

b) This is a binomial distribtuion with n=10, p=0.3

$P(X=3)=C^{10}_3(0.3)^3(0.7)^7=0.267$

c) There is a negative binomial distribution with p=0.3, r=3,x=8.

$P(X=8)=C^{8-1}_{3-1}(1-p)^{x-r}p^r=C^7_2\times0.7^50.3^3=0.0953$

d) This is a binomial distribtuion with n=100, p=0.3

Let's approximate this to normal distribution as n is large.

$\mu=np=100(0.3)=30 \\\sigma^2=np(1-p)=100(0.3)(0.7)=21 \\ \sigma=\sqrt{21}$

We need to apply continuity correction as we are approximating it to normal distribution.

$P(X>8)=P(X>8.5)=1-P(X\le8.5) \\=1-P(z\le\dfrac{8.5-30}{\sqrt{21}}) \\=1-P(z\le-4.69) \\=1-[1-P(z\ge4.69)] \\=P(z\ge4.69) \\=1-P(z<4.69) \\=1-1 \\=0$