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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.
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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. $\newline$ Answer:

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Find probabilities using the binomial distribution

Full solution

Q. 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.

\newline

Answer:

Understand and Determine Formula: Understand the problem and determine the formula to use. $\newline$ We need to find the probability of it raining on exactly one day out of six. This is a binomial probability problem, where $n = 6$ (number of trials), $k = 1$ (number of successes), $p = 0.64$ (probability of success on a single trial), and $q = 1 - p = 0.36$ (probability of failure on a single trial). $\newline$ The formula for the binomial probability is $P(X = k) = \binom{n}{k} \cdot p^k \cdot q^{(n-k)}$ .
Calculate Binomial Coefficient: Calculate the binomial coefficient $(n \choose k)$ for $n = 6$ and $k = 1$ . $\newline$ $(n \choose k) = \frac{n!}{(k! \cdot (n - k)!)}$ $\newline$ $(6 \choose 1) = \frac{6!}{(1! \cdot (6 - 1)!)}$ $\newline$ $(6 \choose 1) = \frac{6}{1}$ $\newline$ $(6 \choose 1) = 6$
Calculate Probabilities: Calculate the probability of success raised to the power of $k$ ( $p^k$ ) and the probability of failure raised to the power of $n-k$ ( $q^{n-k}$ ). $\newline$ $p^k = 0.64^1 = 0.64$ $\newline$ $q^{n-k} = 0.36^{6-1} = 0.36^5$
Calculate $q^{n-k}$ : Calculate $q^{n-k}$ using a calculator. $\newline$ $0.36^5 = 0.010077696$
Multiply Results for P(X = $1$ ): Multiply the results of steps $2$ , $3$ , and $4$ to find the probability $P(X = 1)$ . $P(X = 1) = \binom{6}{1} \times p^1 \times q^5$ $P(X = 1) = 6 \times 0.64 \times 0.010077696$
Perform Final Calculation: Perform the final calculation. $\newline$ $P(X = 1) = 6 \times 0.64 \times 0.010077696$ $\newline$ $P(X = 1) = 0.03877132864$
Round to Nearest Thousandth: Round the answer to the nearest thousandth as requested. $\newline$ $P(X = 1) \approx 0.039$

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