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Naomi and her teammates are each chewing a piece of tropical flavored gum before their championship softball game. $85\%$ of the pieces are pineapple flavored. $\newline$ If $4$ of her teammates are chosen at random, what is the probability that $0$ are chewing pineapple gum? $\newline$ Write your answer as a decimal rounded to the nearest thousandth. $\newline$ ____ $\newline$

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Find probabilities using the binomial distribution

Full solution

Q. Naomi and her teammates are each chewing a piece of tropical flavored gum before their championship softball game.

85\%

of the pieces are pineapple flavored.

\newline

If

4

of her teammates are chosen at random, what is the probability that

0

are chewing pineapple gum?

\newline

Write your answer as a decimal rounded to the nearest thousandth.

\newline

____

\newline

Use Binomial Probability Formula: Use the binomial probability formula $P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}$ . Here, $n = 4$ (number of teammates), $k = 0$ (number of teammates chewing pineapple gum), and $p = 0.85$ (probability of pineapple gum).
Calculate $C(4, 0)$ : Calculate $C(4, 0)$ which is the number of ways to choose $0$ teammates out of $4$ . $\newline$ $C(4, 0) = \frac{4!}{0! \times (4 - 0)!} = 1$ .
Calculate $(0.85)^0$ : Calculate $(0.85)^0$ which is the probability that $0$ teammates are chewing pineapple gum. $\newline$ $(0.85)^0 = 1$ .
Calculate $(1 - 0.85)^{(4 - 0)}$ : Calculate $(1 - 0.85)^{(4 - 0)}$ which is the probability that $4$ teammates are not chewing pineapple gum. $\newline$ $(1 - 0.85)^{(4 - 0)} = (0.15)^4$ .
Multiply Values Together: Multiply all the values together to find the probability. $\newline$ $P(X = 0) = 1 \times 1 \times (0.15)^4$ .
Solve $(0.15)^4$ : Solve $(0.15)^4$ . $(0.15)^4 = 0.15 \times 0.15 \times 0.15 \times 0.15 = 0.00050625$ .
Write Answer as Decimal: Write the answer as a decimal rounded to the nearest thousandth. $P(X = 0) = 0.00050625$ rounded to $0.001$ .

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