If the probability that the Islanders will beat the Rangers in a game is 0.71 , what is the probability that the Islanders will win at most two out of seven games in a series against the Rangers? Round your answer to the nearest thousandth. Answer:

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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) * (p)^k * (1-p)^(n-k), where 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 (seven games in a series), p = 0.71 (probability that the Islanders will beat the Rangers in a game), and k will take on the values 0, 1, and 2 because we are looking for the probability of the Islanders winning at most two games.Calculate Probability for k = 0: Calculate the probability for k = 0. Using the binomial probability formula, we calculate P(X = 0): P(X = 0) = C(7, 0) * (0.71)^0 * (1-0.71)^(7-0) C(7, 0) is the number of ways to choose 0 games out of 7, which is 1. (0.71)^0 is 1 because any number to the power of 0 is 1. (1-0.71)^(7-0) is (0.29)^7. Now we calculate (0.29)^7.Calculate (0.29)^7: Calculate (0.29)^7. (0.29)^7 = 0.29 * 0.29 * 0.29 * 0.29 * 0.29 * 0.29 * 0.29 This calculation gives us approximately 0.0005741. Now we can calculate P(X = 0): P(X = 0) = 1 * 1 * 0.0005741 = 0.0005741.Calculate Probability for k = 1: Calculate the probability for k = 1. Using the binomial probability formula, we calculate P(X = 1): P(X = 1) = C(7, 1) * (0.71)^1 * (1-0.71)^(7-1) C(7, 1) is the number of ways to choose 1 game out of 7, which is 7. (0.71)^1 is 0.71. (1-0.71)^(7-1) is (0.29)^6. Now we calculate (0.29)^6.Calculate (0.29)^6: Calculate (0.29)^6. (0.29)^6 = 0.29 * 0.29 * 0.29 * 0.29 * 0.29 * 0.29 This calculation gives us approximately 0.001979. Now we can calculate P(X = 1): P(X = 1) = 7 * 0.71 * 0.001979 = 0.00994049.Calculate Probability for k = 2: Calculate the probability for k = 2. Using the binomial probability formula, we calculate P(X = 2): P(X = 2) = C(7, 2) * (0.71)^2 * (1-0.71)^(7-2) C(7, 2) is the number of ways to choose 2 games out of 7, which is 7! / (2! * (7-2)!) = 21. (0.71)^2 is approximately 0.5041. (1-0.71)^(7-2) is (0.29)^5. Now we calculate (0.29)^5.Calculate (0.29)^5: Calculate (0.29)^5. (0.29)^5 = 0.29 * 0.29 * 0.29 * 0.29 * 0.29 This calculation gives us approximately 0.006408. Now we can calculate P(X = 2): P(X = 2) = 21 * 0.5041 * 0.006408 = 0.0679057688.Calculate Total Probability: Calculate the total probability for k = 0, k = 1, and k = 2. The total probability that the Islanders will win at most two out of seven games is the sum of the probabilities for k = 0, k = 1, and k = 2: P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) P(X ≤ 2) = 0.0005741 + 0.00994049 + 0.0679057688 P(X ≤ 2) = 0.0784203588 Round this to the nearest thousandth to get the final answer.

If the probability that the Islanders will beat the Rangers in a game is $0$ . $71$ , what is the probability that the Islanders will win at most two out of seven games in a series against the Rangers? Round your answer to the nearest thousandth. $\newline$ Answer:

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If the probability that the Islanders will beat the Rangers in a game is $0$ . $71$ , what is the probability that the Islanders will win at most two out of seven games in a series against the Rangers? Round your answer to the nearest thousandth. $\newline$ Answer: