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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 $31 \%$ . 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. $\newline$ 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

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.

\newline

Answer:

Identify Given Probability: Identify the given probability and the event we are interested in. $\newline$ The probability of rain on any given day is $31\%$ , or $0.31$ 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.
Use Binomial Probability Formula: Use the binomial probability formula to calculate the probability of exactly three days of rain. The binomial probability formula is $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$ , where: - $P(X=k)$ is the probability of $k$ successes in $n$ trials, - $\binom{n}{k}$ is the binomial coefficient, which calculates the number of ways to choose $k$ successes in $n$ trials, - $p$ is the probability of success on a single trial, and - $(1-p)$ is the probability of failure on a single trial. In this case, $n=4$ (four days), $P(X=k)$ $0$ (three days of rain), and $P(X=k)$ $1$ (probability of rain on any given day).
Calculate Binomial Coefficient: Calculate the binomial coefficient $\binom{4}{3}$ . $\binom{4}{3} = \frac{4!}{3! \cdot (4-3)!} = \frac{4}{1} = 4$ There are $4$ ways to have exactly three days of rain in four days.
Calculate Probability of Three Days: Calculate the probability of exactly three days of rain using the binomial probability formula. $\newline$ $P(X=3) = \binom{4}{3} \times 0.31^3 \times (1-0.31)^{4-3}$ $\newline$ $P(X=3) = 4 \times 0.31^3 \times 0.69^1$
Perform Calculations: Perform the calculations. $\newline$ $P(X=3) = 4 \times (0.31 \times 0.31 \times 0.31) \times 0.69$ $\newline$ $P(X=3) = 4 \times 0.029791 \times 0.69$ $\newline$ $P(X=3) = 4 \times 0.02055579$ $\newline$ $P(X=3) = 0.08222316$
Round Answer: Round the answer to the nearest thousandth as requested. $P(X=3) \approx 0.082$

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