The Hiking Club plans to go camping in a State park where the probability of rain on any given day is 83% . What is the probability that it will rain on exactly zero of the five days they are there?Round your answer to the nearest thousandth.

To solve this problem, we can use the binomial probability formula. The binomial probability formula is given by:<br /><br />\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]<br /><br />where:<br />- \( n \) is the number of trials (in this case, 5 days),<br />- \( k \) is the number of successes (in this case, 0 days of rain),<br />- \( p \) is the probability of success on a single trial (in this case, 0.83 for rain),<br />- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).<br /><br />Given:<br />- \( n = 5 \)<br />- \( k = 0 \)<br />- \( p = 0.83 \)<br /><br />First, calculate the binomial coefficient \( \binom{5}{0} \):<br /><br />\[ \binom{5}{0} = \frac{5!}{0!(5-0)!} = \frac{5!}{0! \cdot 5!} = 1 \]<br /><br />Next, calculate \( p^k \) and \( (1-p)^{n-k} \):<br /><br />\[ p^k = 0.83^0 = 1 \]<br />\[ (1-p)^{n-k} = (1-0.83)^5 = 0.17^5 \]<br /><br />Now, compute \( 0.17^5 \):<br /><br />\[ 0.17^5 \approx 0.0001419857 \]<br /><br />Finally, multiply these values together to get the probability:<br /><br />\[ P(X = 0) = \binom{5}{0} \cdot 0.83^0 \cdot 0.17^5 \]<br />\[ P(X = 0) = 1 \cdot 1 \cdot 0.0001419857 \]<br />\[ P(X = 0) \approx 0.000142 \]<br /><br />Rounding to the nearest thousandth, the probability that it will rain on exactly zero of the five days they are there is approximately \( 0.000 \).