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Lauren created her own deck of cards for a magic trick. She decorated all the cards with pictures of her family members. $12\%$ of the cards contained pictures of her brother. $\newline$ If Lauren draws a random card from the deck $2$ times, what is the probability that exactly $2$ of the cards have a picture of Lauren's brother? $\newline$ 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. Lauren created her own deck of cards for a magic trick. She decorated all the cards with pictures of her family members.

12\%

of the cards contained pictures of her brother.

\newline

If Lauren draws a random card from the deck

2

times, what is the probability that exactly

2

of the cards have a picture of Lauren's brother?

\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 $p = 0.12$ .
Calculate $C(2, 2)$ : Calculate $C(2, 2)$ which is $\frac{2!}{2!(2 - 2)!}$ . That simplifies to $1$ .
Solve $(0.12)^2$ : Solve $(0.12)^2$ which is $0.12 \times 0.12$ . That equals $0.0144$ .
Simplify $(1 - 0.12)^{(2 - 2)}$ : Simplify $(1 - 0.12)^{(2 - 2)}$ which is $(0.88)^0$ . Anything to the power of $0$ is $1$ .
Multiply Values Together: Now multiply all the values together: $P(X = 2) = 1 \times 0.0144 \times 1$ . That equals $0.0144$ .
Round to Nearest Thousandth: Round the answer to the nearest thousandth: $0.0144$ rounds to $0.014$ .

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