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The Hiking Club plans to go camping in a State park where the probability of rain on any given day is 0.64 . What is the probability that it will rain on exactly one of the six days they are there? Round your answer to the nearest thousandth. Answer:

The Hiking Club plans to go camping in a State park where the probability of rain on any given day is 00.6464 . What is the probability that it will rain on exactly one of the six days they are there? Round your answer to the nearest thousandth.\newlineAnswer:

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Q. The Hiking Club plans to go camping in a State park where the probability of rain on any given day is 00.6464 . What is the probability that it will rain on exactly one of the six days they are there? Round your answer to the nearest thousandth.\newlineAnswer:
  1. Understand and Determine Formula: Understand the problem and determine the formula to use.\newlineWe need to find the probability of it raining on exactly one day out of six. This is a binomial probability problem, where n=6n = 6 (number of trials), k=1k = 1 (number of successes), p=0.64p = 0.64 (probability of success on a single trial), and q=1−p=0.36q = 1 - p = 0.36 (probability of failure on a single trial).\newlineThe formula for the binomial probability is P(X=k)=(nk)⋅pk⋅q(n−k)P(X = k) = \binom{n}{k} \cdot p^k \cdot q^{(n-k)}.
  2. Calculate Binomial Coefficient: Calculate the binomial coefficient ((nk))(n \choose k) for n=6n = 6 and k=1k = 1.\newline((nk)=n!(k!⋅(n−k)!))(n \choose k) = \frac{n!}{(k! \cdot (n - k)!)}\newline((61)=6!(1!⋅(6−1)!))(6 \choose 1) = \frac{6!}{(1! \cdot (6 - 1)!)}\newline((61)=61)(6 \choose 1) = \frac{6}{1}\newline((61)=6)(6 \choose 1) = 6
  3. Calculate Probabilities: Calculate the probability of success raised to the power of kk (pkp^k) and the probability of failure raised to the power of n−kn-k (qn−kq^{n-k}).\newlinepk=0.641=0.64p^k = 0.64^1 = 0.64\newlineqn−k=0.366−1=0.365q^{n-k} = 0.36^{6-1} = 0.36^5
  4. Calculate qn−kq^{n-k}: Calculate qn−kq^{n-k} using a calculator.\newline0.365=0.0100776960.36^5 = 0.010077696
  5. Multiply Results for P(X = 11): Multiply the results of steps 22, 33, and 44 to find the probability P(X=1)P(X = 1).P(X=1)=(61)×p1×q5P(X = 1) = \binom{6}{1} \times p^1 \times q^5P(X=1)=6×0.64×0.010077696P(X = 1) = 6 \times 0.64 \times 0.010077696
  6. Perform Final Calculation: Perform the final calculation.\newlineP(X=1)=6×0.64×0.010077696P(X = 1) = 6 \times 0.64 \times 0.010077696\newlineP(X=1)=0.03877132864P(X = 1) = 0.03877132864
  7. Round to Nearest Thousandth: Round the answer to the nearest thousandth as requested.\newlineP(X=1)≈0.039P(X = 1) \approx 0.039

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