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Sophia and her teammates are each chewing a piece of tropical flavored gum before their championship softball game. $58\%$ of the pieces are pineapple flavored. $\newline$ If $2$ of her teammates are chosen at random, what is the probability that exactly $2$ 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. Sophia and her teammates are each chewing a piece of tropical flavored gum before their championship softball game.

58\%

of the pieces are pineapple flavored.

\newline

If

2

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

2

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, $n = 2$ , $k = 2$ , and $p = 0.58$ .
Calculate Combination: Calculate $C(2, 2) = \frac{2!}{2!(2 - 2)!} = 1$ .
Calculate $(0.58)^2$ : Solve $(0.58)^2 = 0.58 \times 0.58 = 0.3364$ .
Simplify $(1 - 0.58)^{(2 - 2)}$ : Simplify $(1 - 0.58)^{(2 - 2)} = (0.42)^0 = 1$ .
Multiply Values and Round: Multiply all the values together: $P(X = 2) = 1 \times 0.3364 \times 1 = 0.3364$ . Round to the nearest thousandth: $0.336$ .

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