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19%19\% of students in a class have black hair. If 44 students are chosen at random, what is the probability that 00 have black hair? Write your answer as a decimal rounded to the nearest thousandth. ____

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Q. 19%19\% of students in a class have black hair. If 44 students are chosen at random, what is the probability that 00 have black hair? 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=4n = 4, k=0k = 0, and p=0.19p = 0.19.
  2. Calculate C(4,0)C(4, 0): Calculate C(4,0)C(4, 0) which is 4!0!(4−0)!\frac{4!}{0! (4 - 0)!}. Since 0!0! is 11 and n!n!\frac{n!}{n!} is 11, C(4,0)C(4, 0) equals 11.
  3. Solve (0.19)0(0.19)^0: Solve (0.19)0(0.19)^0. Any number to the power of 00 is 11, so (0.19)0(0.19)^0 equals 11.
  4. Simplify (1−0.19)(4−0)(1 - 0.19)^{(4 - 0)}: Simplify (1−0.19)(4−0)(1 - 0.19)^{(4 - 0)}. This is (0.81)4(0.81)^4.
  5. Calculate (0.81)4(0.81)^4: Calculate (0.81)4(0.81)^4. This equals 0.81×0.81×0.81×0.810.81 \times 0.81 \times 0.81 \times 0.81.
  6. Final Calculation: 0.81×0.81×0.81×0.810.81 \times 0.81 \times 0.81 \times 0.81 equals 0.430467210.43046721.

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