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Mrs. Gomes found that 40% of students at her high school take chemistry. She randomly surveys 12 students. What is the probability that exactly 4 students have taken chemistry? Round the answer to the nearest thousandth.

Mrs. Gomes found that 40% 40 \% of students at her high school take chemistry. She randomly surveys 1212 students. What is the probability that exactly 44 students have taken chemistry? Round the answer to the nearest thousandth.

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Full solution

Q. Mrs. Gomes found that 40% 40 \% of students at her high school take chemistry. She randomly surveys 1212 students. What is the probability that exactly 44 students have taken chemistry? Round the answer to the nearest thousandth.
  1. Identify values: Identify the values of nn, kk, and pp for the binomial probability formula. Here, n=12n = 12 (number of students surveyed), k=4k = 4 (students who have taken chemistry), and p=0.40p = 0.40 (probability of a student taking chemistry).
  2. Use 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)}. Substitute the values: P(X=4)=C(12,4)⋅(0.40)4⋅(0.60)(12−4)P(X = 4) = C(12, 4) \cdot (0.40)^4 \cdot (0.60)^{(12-4)}.
  3. Calculate C(12,4)C(12, 4): Calculate C(12,4)C(12, 4), which is the number of combinations of 1212 students taken 44 at a time. C(12,4)=12!4!×(12−4)!=495C(12, 4) = \frac{12!}{4! \times (12-4)!} = 495.
  4. Calculate (0.40)4(0.40)^4: Calculate (0.40)4(0.40)^4. (0.40)4=0.40×0.40×0.40×0.40=0.0256(0.40)^4 = 0.40 \times 0.40 \times 0.40 \times 0.40 = 0.0256.
  5. Calculate (0.60)8(0.60)^8: Calculate (0.60)8(0.60)^8. (0.60)8=0.60×0.60×0.60×0.60×0.60×0.60×0.60×0.60=0.016777216(0.60)^8 = 0.60 \times 0.60 \times 0.60 \times 0.60 \times 0.60 \times 0.60 \times 0.60 \times 0.60 = 0.016777216.
  6. Multiply values: Multiply all the values together to find the probability. P(X=4)=495×0.0256×0.016777216=0.21233664P(X = 4) = 495 \times 0.0256 \times 0.016777216 = 0.21233664.

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