Question

geometric random variables 71% of the surface of the earth is covered in water. a random number generator uses latitude and longitude to select a random location on earth. if such locations are generated, what is the probability that the first of those locations that is over land is on the 8th location? $(0.29)^8$ $(0.71)^8$ $(0.71)^7(0.29)$ $(0.29)^7(0.71)$

Explanation:

Step1: Understand the problem

We need to find the probability that the first location over land is the 8th location. This means the first 7 locations are over water, and the 8th is over land.

Step2: Determine probabilities

The probability of a location being over water is \( 0.71 \) (since 71% is water), and the probability of a location being over land is \( 0.29 \) (since \( 1 - 0.71 = 0.29 \)).

Step3: Apply geometric probability formula

For a geometric distribution, the probability that the first success (here, land) occurs on the \( n \)-th trial is \( (1 - p)^{n - 1}p \), where \( p \) is the probability of success. Here, \( n = 8 \), \( p = 0.29 \), and \( 1 - p = 0.71 \). So the probability is \( (0.71)^{7}(0.29) \).

Answer:

\( (0.71)^{7}(0.29) \) (the third option)