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This Friday, Tucker has a French vocabulary test as well as a Spanish vocabulary test. To prepare, he made a study card for each word, $44\%$ of which are French. Every time he picks a card, he sticks it back in the deck and shuffles again. $\newline$ If Tucker picks a study card from the deck $5$ times during his first study session, what is the probability that exactly $3$ cards have a French word? $\newline$ Write your answer as a decimal rounded to the nearest thousandth. $\newline$ ____ $\newline$

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Find probabilities using the binomial distribution

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Q. This Friday, Tucker has a French vocabulary test as well as a Spanish vocabulary test. To prepare, he made a study card for each word,

44\%

of which are French. Every time he picks a card, he sticks it back in the deck and shuffles again.

\newline

If Tucker picks a study card from the deck

5

times during his first study session, what is the probability that exactly

3

cards have a French word?

\newline

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

\newline

____

\newline

Find French Card Probability: First, we need to find the probability of picking a French card. Since $44\%$ of the cards are French, the probability of picking a French card is $0.44$ .
Calculate Non-French Card Probability: The probability of picking a non-French card is then $1 - 0.44$ , which is $0.56$ .
Apply Binomial Probability Formula: We use the binomial probability formula, which is $P(X=k) = \binom{n}{k} \cdot (p^k) \cdot ((1-p)^{(n-k)})$ , where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $\binom{n}{k}$ is the binomial coefficient.
Calculate $(5 \choose 3)$ : For exactly $3$ French cards, we have $n=5$ , $k=3$ , and $p=0.44$ . So we need to calculate $(5 \choose 3) \times (0.44^3) \times (0.56^{(5-3)})$ .
Calculate $0.44^3$ : Calculating $(5 \choose 3)$ gives us $\frac{5!}{3! \cdot (5-3)!} = 10$ .
Calculate $0.56^2$ : Now we calculate $0.44^3$ which is $0.085184$ .
Multiply Results: Next, we calculate $0.56^2$ which is $0.3136$ .
Multiply Results: Next, we calculate $0.56^2$ which is $0.3136$ . Multiplying these together, we get $10 \times 0.085184 \times 0.3136 = 0.2673$ .

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