Answer to Question #195486 in Statistics and Probability for Mohammed Moro

Question

An oil prospector will drill a succession of holes in a given area to find a productive well.The probability that he is successful on a given trial is 0.02.

(a). What is the probability that the third hole drilled is the first to yield productive well?

(b). If the prospector can afford to drill at most ten wells,what is the probability that he will fail to find a productive well?

Answer 1

(a). Let Y be the number of trial on which the prospector will find the first productive well.

Since Y is a geometric random variable with $p=0.02$

The probability that the third hole drilled is the first to yield a productive well is:

$P\left(Y=3\right)=q^2\times p=\left(1-0.02\right)^2\left(0.02\right)=0.0192$

(b). Y is a binomial random variable with $n=10$ and $p=0.02$

$q=1-p=1-0.02=0.98$

Geometric probability is defined as:

$p\left(y\right)=q^{y-1}p$

Then the corresponding probabilities are added as follows:

$p\left(Y\le 10\right)=p\left(0\right)+.....+p\left(10\right)=0.8926$

Finally, the complement rule is used to get:

$p\left(Y>10\right)=1-P\left(Y\le 10\right)=1-0.8926=0.1074=10.74\%$ .