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Maura created her own deck of cards for a magic trick. She decorated all the cards with pictures of her family members. 16%16\% of the cards contained pictures of her brother. If Maura draws a random card from the deck 22 times, what is the probability that exactly 11 of the cards has a picture of Maura's brother? Write your answer as a decimal rounded to the nearest thousandth.____

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Q. Maura created her own deck of cards for a magic trick. She decorated all the cards with pictures of her family members. 16%16\% of the cards contained pictures of her brother. If Maura draws a random card from the deck 22 times, what is the probability that exactly 11 of the cards has a picture of Maura's brother? 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=2n = 2, k=1k = 1, and p=0.16p = 0.16.
  2. Calculate C(2,1)C(2, 1): Calculate C(2,1)C(2, 1) which is 2!1!×(2−1)!\frac{2!}{1! \times (2 - 1)!}. This equals 21×1\frac{2}{1 \times 1} which is 22.
  3. Calculate (0.16)1(0.16)^1: Now calculate (0.16)1(0.16)^1 which is just 0.160.16.
  4. Calculate (1−0.16)(2−1)(1 - 0.16)^{(2 - 1)}: Calculate (1−0.16)(2−1)(1 - 0.16)^{(2 - 1)} which is (0.84)1(0.84)^1 and that equals 0.840.84.
  5. Multiply values together: Multiply all the values together: P(X=1)=2×0.16×0.84P(X = 1) = 2 \times 0.16 \times 0.84. This equals 0.26880.2688.
  6. Round to nearest thousandth: Round the answer to the nearest thousandth: 0.26880.2688 rounded is 0.2690.269.

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