The Hiking Club plans to go camping in a State park where the probability of rain on any given day is 1/4 . What is the probability that it will rain on exactly one 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 used to find the probability of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials.<br /><br />The binomial probability formula is given by:<br /><br />P(X = k) = (n choose k) * p^k * (1-p)^(n-k)<br /><br />Where:<br />- P(X = k) is the probability of obtaining k successes in n trials<br />- n is the number of trials<br />- k is the number of successes<br />- p is the probability of success in a single trial<br />- (1-p) is the probability of failure in a single trial<br /><br />In this case, the Hiking Club plans to go camping in park of rain on any given day is 1/4. We want to find the probability that it will rain on exactly one of the five days they are there.<br /><br />Given information:<br />- Probability of rain on any given day (p) = 1/4<br />- Number of days they are there (n) = 5<br />- Number of days it will rain (k) = 1<br /><br />Plugging these values into the binomial probability formula:<br /><br />P(X = 1) = (5 choose 1) * (1/4)^1 * (3/4)^(5-1)<br />P(X = 1) = 5 * (1/4) * (3/4)^4<br />P(X = 1) = 5 * (1/4) * (81/256)<br />P(X = 1) = 5 * (1/4) * (81/256)<br />P(X = 1) = 5 * (81/1024)<br />P(X = 1) = 405/1024<br />P(X = 1) ≈ 0.396<br /><br />Therefore, the probability that it will rain on exactly one of the five days they are there is approximately 0.396, rounded to the nearest thousandth.