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If the probability that the Islanders will beat the Rangers in a game is
61%, what is the probability that the Islanders will win exactly five out of seven 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 $61 \%$ , what is the probability that the Islanders will win exactly five out of seven 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

61 \%

, what is the probability that the Islanders will win exactly five out of seven 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 $P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}$ . Here, $n$ is the number of trials, $k$ is the number of successes, and $p$ is the probability of success on a single trial. For this problem: $n = 7$ (since there are seven games), $k = 5$ (since we want to find the probability of exactly five wins), $p = 0.61$ (since the probability of the Islanders winning a single game is $61$ %).
Calculate binomial coefficient: Calculate the binomial coefficient $C(n, k)$ . The binomial coefficient $C(n, k)$ is calculated using the formula $\frac{n!}{k!(n - k)!}$ . Substitute $n = 7$ and $k = 5$ into the formula: $C(7, 5) = \frac{7!}{5!(7 - 5)!}$ = $\frac{7 \times 6}{2 \times 1}$ = $\frac{42}{2}$ = $21$ .
Calculate probability of five wins: Calculate the probability of exactly five wins $P(X = 5)$ . Using the binomial probability formula: $P(X = 5) = C(7, 5) \cdot (0.61)^5 \cdot (1 - 0.61)^{(7 - 5)}$ Substitute the values we have calculated: $P(X = 5) = 21 \cdot (0.61)^5 \cdot (0.39)^2$ .
Solve probabilities: Solve $(0.61)^5$ and $(0.39)^2$ . $\newline$ Calculate $(0.61)^5$ : $\newline$ $(0.61)^5 = 0.61 \times 0.61 \times 0.61 \times 0.61 \times 0.61$ $\newline$ $\approx 0.088567$ . $\newline$ Calculate $(0.39)^2$ : $\newline$ $(0.39)^2 = 0.39 \times 0.39$ $\newline$ $\approx 0.1521$ .
Multiply values for probability: Multiply all the values together to find the probability $P(X = 5)$ . $P(X = 5) = 21 \times 0.088567 \times 0.1521 \approx 0.282711 \times 0.1521 \approx 0.043004.$
Round to nearest thousandth: Round the answer to the nearest thousandth. $\newline$ The probability $P(X = 5)$ rounded to the nearest thousandth is: $\newline$ $0.043$ .

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