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Last week, a flower delivery business conducted a survey on customer happiness. They found that 56%56\% of their customers were ＂pleased＂ with the service.\newlineIf the company surveys 44 of their customers the next week, what is the probability that exactly 11 is pleased?\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____\newline

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Q. Last week, a flower delivery business conducted a survey on customer happiness. They found that 56%56\% of their customers were ＂pleased＂ with the service.\newlineIf the company surveys 44 of their customers the next week, what is the probability that exactly 11 is pleased?\newlineWrite your answer as a decimal rounded to the nearest thousandth.\newline____\newline
  1. Use Binomial Probability Formula: Use the binomial probability formula: P(X=k)=C(n,k)⋅(p)k⋅(1−p)(n−k)P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}. Here, n=4n = 4, k=1k = 1, and p=0.56p = 0.56.
  2. Calculate C(4,1)C(4, 1): Calculate C(4,1)C(4, 1) using the formula n!k!(n−k)!\frac{n!}{k!(n - k)!}. So, C(4,1)=4!1!×(4−1)!=41=4C(4, 1) = \frac{4!}{1! \times (4 - 1)!} = \frac{4}{1} = 4.
  3. Compute (0.56)1(0.56)^1: Compute (0.56)1(0.56)^1 which is just 0.560.56.
  4. Calculate (1−0.56)(4−1)(1 - 0.56)^{(4 - 1)}: Calculate (1−0.56)(4−1)(1 - 0.56)^{(4 - 1)} which is (0.44)3(0.44)^3. So, (0.44)3=0.44×0.44×0.44=0.085184(0.44)^3 = 0.44 \times 0.44 \times 0.44 = 0.085184.
  5. Multiply Values Together: Multiply all the values together: P(X=1)=4×0.56×0.085184P(X = 1) = 4 \times 0.56 \times 0.085184. So, P(X=1)=4×0.56×0.085184=0.1905664P(X = 1) = 4 \times 0.56 \times 0.085184 = 0.1905664.
  6. Round to Nearest Thousandth: Round the answer to the nearest thousandth: 0.19056640.1905664 rounds to 0.1910.191.

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