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A certain insecticide kills 60%60\% of all insects in laboratory experiments. A sample of 1212 insects is exposed to the insecticide in a particular experiment. What is the probability that exactly 1010 insects will die?

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Full solution

Q. A certain insecticide kills 60%60\% of all insects in laboratory experiments. A sample of 1212 insects is exposed to the insecticide in a particular experiment. What is the probability that exactly 1010 insects will die?
  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)}\newlineHere, n=12n = 12, k=10k = 10, and p=0.60p = 0.60.
  2. Calculate C(12,10)C(12, 10): Calculate C(12,10)C(12, 10) using the formula C(n,k)=n!k!(n−k)!C(n, k) = \frac{n!}{k!(n - k)!}.C(12,10)=12!(10!∗(12−10)!)=12!(10!∗2!)=(12∗11)(2∗1)=66C(12, 10) = \frac{12!}{(10! * (12 - 10)!)} = \frac{12!}{(10! * 2!)} = \frac{(12 * 11)}{(2 * 1)} = 66.
  3. Calculate (0.60)10(0.60)^{10}: Calculate (0.60)10(0.60)^{10}.(0.60)10=0.60×0.60×0.60×0.60×0.60×0.60×0.60×0.60×0.60×0.60≈0.0060466176(0.60)^{10} = 0.60 \times 0.60 \times 0.60 \times 0.60 \times 0.60 \times 0.60 \times 0.60 \times 0.60 \times 0.60 \times 0.60 \approx 0.0060466176.
  4. Calculate (1−0.60)(12−10)(1 - 0.60)^{(12 - 10)}: Calculate (1−0.60)(12−10)(1 - 0.60)^{(12 - 10)}.\newline(1−0.60)(12−10)=(0.40)2=0.40×0.40=0.16(1 - 0.60)^{(12 - 10)} = (0.40)^2 = 0.40 \times 0.40 = 0.16.
  5. Multiply Values to Find Probability: Multiply all the values together to find the probability.\newlineP(X=10)=66×0.0060466176×0.16≈0.063848P(X = 10) = 66 \times 0.0060466176 \times 0.16 \approx 0.063848.

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