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Colleen and her teammates are each chewing a piece of tropical flavored gum before their championship softball game. 40%40\% of the pieces are pineapple flavored. If 22 of her teammates are chosen at random, what is the probability that 00 are chewing pineapple gum? Write your answer as a decimal rounded to the nearest thousandth. ____

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Q. Colleen and her teammates are each chewing a piece of tropical flavored gum before their championship softball game. 40%40\% of the pieces are pineapple flavored. If 22 of her teammates are chosen at random, what is the probability that 00 are chewing pineapple gum? 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=0k = 0, and p=0.40p = 0.40 for pineapple flavor.
  2. Calculate C(2,0)C(2, 0): Calculate C(2,0)C(2, 0) which is the number of ways to choose 00 pineapple gums from 22.\newlineC(2,0)=2!0!×(2−0)!=1C(2, 0) = \frac{2!}{0! \times (2 - 0)!} = 1.
  3. Compute (0.40)0(0.40)^0: Compute (0.40)0(0.40)^0 which is the probability of choosing 00 pineapple gums.\newline(0.40)0=1(0.40)^0 = 1.
  4. Calculate (1−0.40)(2−0)(1 - 0.40)^{(2 - 0)}: Calculate (1−0.40)(2−0)(1 - 0.40)^{(2 - 0)} which is the probability of not choosing pineapple gums twice.\newline(1−0.40)(2−0)=(0.60)2=0.36(1 - 0.40)^{(2 - 0)} = (0.60)^2 = 0.36.
  5. Multiply Values to Find Probability: Multiply all the values together to find the probability.\newlineP(X=0)=1×1×0.36=0.36P(X = 0) = 1 \times 1 \times 0.36 = 0.36.

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