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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 31%. What is the probability that it will rain on exactly three of the four 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 31% 31 \% . What is the probability that it will rain on exactly three of the four 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 31% 31 \% . What is the probability that it will rain on exactly three of the four days they are there? Round your answer to the nearest thousandth.\newlineAnswer:
  1. Identify Given Probability: Identify the given probability and the event we are interested in.\newlineThe probability of rain on any given day is 31%31\%, or 0.310.31 when expressed as a decimal. We want to find the probability that it will rain on exactly three out of the four days they are camping.
  2. Use Binomial Probability Formula: Use the binomial probability formula to calculate the probability of exactly three days of rain. The binomial probability formula is P(X=k)=(nk)⋅pk⋅(1−p)n−kP(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}, where: - P(X=k)P(X=k) is the probability of kk successes in nn trials, - (nk)\binom{n}{k} is the binomial coefficient, which calculates the number of ways to choose kk successes in nn trials, - pp is the probability of success on a single trial, and - (1−p)(1-p) is the probability of failure on a single trial. In this case, n=4n=4 (four days), P(X=k)P(X=k)00 (three days of rain), and P(X=k)P(X=k)11 (probability of rain on any given day).
  3. Calculate Binomial Coefficient: Calculate the binomial coefficient (43)\binom{4}{3}.(43)=4!3!⋅(4−3)!=41=4\binom{4}{3} = \frac{4!}{3! \cdot (4-3)!} = \frac{4}{1} = 4There are 44 ways to have exactly three days of rain in four days.
  4. Calculate Probability of Three Days: Calculate the probability of exactly three days of rain using the binomial probability formula.\newlineP(X=3)=(43)×0.313×(1−0.31)4−3P(X=3) = \binom{4}{3} \times 0.31^3 \times (1-0.31)^{4-3}\newlineP(X=3)=4×0.313×0.691P(X=3) = 4 \times 0.31^3 \times 0.69^1
  5. Perform Calculations: Perform the calculations.\newlineP(X=3)=4×(0.31×0.31×0.31)×0.69P(X=3) = 4 \times (0.31 \times 0.31 \times 0.31) \times 0.69\newlineP(X=3)=4×0.029791×0.69P(X=3) = 4 \times 0.029791 \times 0.69\newlineP(X=3)=4×0.02055579P(X=3) = 4 \times 0.02055579\newlineP(X=3)=0.08222316P(X=3) = 0.08222316
  6. Round Answer: Round the answer to the nearest thousandth as requested. P(X=3)≈0.082P(X=3) \approx 0.082

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