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If the probability that the Islanders will beat the Rangers in a game is
18%, what is the probability that the Islanders will win exactly one out of six games in a series against the Rangers? Round your answer to the nearest thousandth.
Answer:

If the probability that the Islanders will beat the Rangers in a game is $18 \%$ , what is the probability that the Islanders will win exactly one out of six games in a series against the Rangers? Round your answer to the nearest thousandth. $\newline$ Answer:

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Find probabilities using the binomial distribution

Full solution

Q. If the probability that the Islanders will beat the Rangers in a game is

18 \%

, what is the probability that the Islanders will win exactly one out of six games in a series against the Rangers? Round your answer to the nearest thousandth.

\newline

Answer:

Identify Formula and Values: Identify the binomial probability formula and the values of $n$ , $k$ , and $p$ . The binomial probability formula is $P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}$ , where: - $n$ is the number of trials, - $k$ is the number of successes, - $p$ is the probability of success on a single trial. For this problem: $n = 6$ (since there are six games), $k = 1$ (since we want to find the probability of exactly one win), $p = 0.18$ (since the probability of the Islanders winning a game is $18$ %).
Calculate Binomial Coefficient: Calculate the binomial coefficient $C(n, k)$ . The binomial coefficient $C(n, k)$ is calculated using the formula $\frac{n!}{k!(n - k)!}$ . For $n = 6$ and $k = 1$ , we have: $C(6, 1) = \frac{6!}{1!(6 - 1)!} = \frac{6}{1 \times 5!} = \frac{6}{120} = \frac{1}{20} = 0.05$ .
Calculate Probability of One Win: Calculate the probability of exactly one win using the binomial probability formula. $\newline$ Substitute $n = 6$ , $k = 1$ , and $p = 0.18$ into the formula $P(X = k) = C(n, k) \cdot (p)^k \cdot (1 - p)^{(n - k)}$ : $\newline$ $P(X = 1) = C(6, 1) \cdot (0.18)^1 \cdot (1 - 0.18)^{(6 - 1)}$ .
Solve Probability Expression: Solve the probability expression. $\newline$ First, calculate $(0.18)^1$ , which is simply $0.18$ . $\newline$ Next, calculate $(1 - 0.18)^{(6 - 1)}$ , which is $(0.82)^5$ . $\newline$ $(0.82)^5 = 0.82 \times 0.82 \times 0.82 \times 0.82 \times 0.82 \approx 0.32890368$ . $\newline$ Now, multiply $C(6, 1)$ , $(0.18)^1$ , and $(0.82)^5$ together: $\newline$ $P(X = 1) = 0.05 \times 0.18 \times 0.32890368 \approx 0.00296553$ .
Round to Nearest Thousandth: Round the answer to the nearest thousandth. $\newline$ $P(X = 1) \approx 0.00296553$ rounded to the nearest thousandth is $0.003$ .

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