To solve this problem, we need to understand the sequence of events that leads to the first land location occurring on the 8th attempt. The process is repeated trials where we are interested in the first success (a land location) occurring after a number of failures (water locations).
Here are the detailed steps to find this probability:
1. Determine the probabilities:
- Probability that a random location is over water (failure) is 0.71 (since 71% of Earth is covered in water).
- Probability that a random location is over land (success) is 0.29 (since 29% of Earth is covered in land).
2. Set up the scenario:
- We seek the probability that the first 7 locations are water and the 8th location is land.
3. Calculate the individual components:
- The probability that a single randomly chosen location is water is 0.71.
- The probability that a single randomly chosen location is land is 0.29.
4. Combine the probabilities:
- For the first 7 locations being over water, the combined probability is [tex]\((0.71)^7\)[/tex].
- For the 8th location being over land, the probability is [tex]\(0.29\)[/tex].
5. Multiply the probabilities:
- The joint probability of the first 7 locations being water and the 8th being land is:
[tex]\[
(0.71)^7 \times 0.29
\][/tex]
6. Substitute and calculate the result:
- Using the values and the multiplication rule of probabilities, we find:
[tex]\[
(0.71)^7 \times 0.29 \approx 0.02637584845933389
\][/tex]
Hence, the probability that the first location over land is on the 8th location is:
[tex]\[
(0.71)^7(0.29)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{(0.71)^7(0.29)}
\][/tex]
