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78%78\% of cows on a farm are pregnant. If 55 cows are chosen at random, what is the probability that exactly 33 are pregnant? Write your answer as a decimal rounded to the nearest thousandth. ____

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Q. 78%78\% of cows on a farm are pregnant. If 55 cows are chosen at random, what is the probability that exactly 33 are pregnant? Write your answer as a decimal rounded to the nearest thousandth. ____
  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=5n = 5, k=3k = 3, and p=0.78p = 0.78.
  2. Calculate C(5,3)C(5, 3): Calculate C(5,3)C(5, 3) using the formula n!k!(n−k)!\frac{n!}{k!(n - k)!}. So, C(5,3)=5!3!(5−3)!=5×42×1=10C(5, 3) = \frac{5!}{3!(5 - 3)!} = \frac{5 \times 4}{2 \times 1} = 10.
  3. Compute (0.78)3(0.78)^3: Compute (0.78)3(0.78)^3 which is 0.78×0.78×0.78=0.4745520.78 \times 0.78 \times 0.78 = 0.474552.
  4. Calculate (1−0.78)(5−3)(1 - 0.78)^{(5 - 3)}: Calculate (1−0.78)(5−3)(1 - 0.78)^{(5 - 3)} which is (0.22)2=0.0484(0.22)^2 = 0.0484.
  5. Multiply Values Together: Now, multiply all the values together: P(X=3)=10×0.474552×0.0484=0.229776768P(X = 3) = 10 \times 0.474552 \times 0.0484 = 0.229776768.
  6. Round to Nearest Thousandth: Round the answer to the nearest thousandth: 0.2297767680.229776768 rounds to 0.2300.230.

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