To solve this problem, we need to understand how to calculate the probability that it will NOT rain on any of the next three days, given that the probability of rain on a particular day is [tex]\(\frac{2}{3}\)[/tex].
1. Identify the probability of no rain on a single day:
- The probability that it rains on a given day is [tex]\(\frac{2}{3}\)[/tex].
- Therefore, the probability that it does NOT rain on a given day is the complement of the probability of rain.
[tex]\[
\text{Probability of no rain in a day} = 1 - \frac{2}{3} = \frac{1}{3}
\][/tex]
2. Calculate the probability that it will NOT rain on any of the next three days:
- We assume the weather events on different days are independent.
- Hence, the probability that it will not rain on any of the three days is the product of the probabilities of no rain on each of the three days.
[tex]\[
\text{Probability of no rain over three days} = \left(\frac{1}{3}\right)^3
\][/tex]
3. Compute [tex]\(\left(\frac{1}{3}\right)^3\)[/tex]:
[tex]\[
\left(\frac{1}{3}\right)^3 = \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}
\][/tex]
Therefore, the probability that it will NOT rain on any of these three days is [tex]\(\frac{1}{27}\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{\frac{1}{27}} \][/tex]
