This Friday, Eva has a French vocabulary test as well as a Spanish vocabulary test. To prepare, she made a study card for each word, 40% of which are French. Every time she picks a card, she sticks it back in the deck and shuffles again. If Eva picks a study card from the deck 4 times during her first study session, what is the probability that exactly 4 cards have a French word? Write your answer as a decimal rounded to the nearest thousandth. ____

Identify values: Identify the values of n, k, and p for the binomial probability formula. n = 4 (number of trials) k = 4 (number of successful French word picks) p = 0.40 (probability of picking a French card)Use binomial formula: Use the binomial probability formula P(X = k) = C(n, k) * (p)^k * (1-p)^(n-k). Substitute n = 4, k = 4, and p = 0.40 into the formula. P(X = 4) = C(4, 4) * (0.40)^4 * (1 - 0.40)^(4 - 4)Calculate C(4, 4): Calculate the value of C(4, 4). C(4, 4) = 4! / 4! (4 - 4)! = 1Solve (0.40)^4: Solve (0.40)^4. (0.40)^4 = 0.40 * 0.40 * 0.40 * 0.40 = 0.0256Simplify (1 - 0.40)^(4 - 4): Simplify (1 - 0.40)^(4 - 4). (1 - 0.40)^(4 - 4) = (0.60)^0 = 1Multiply values to find probability: Multiply all the values together to find the probability. P(X = 4) = 1 * 0.0256 * 1 = 0.0256 Round the answer to the nearest thousandth. P(X = 4) = 0.026

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This Friday, Eva has a French vocabulary test as well as a Spanish vocabulary test. To prepare, she made a study card for each word, $40\%$ of which are French. Every time she picks a card, she sticks it back in the deck and shuffles again. $\newline$ If Eva picks a study card from the deck $4$ times during her first study session, what is the probability that exactly $4$ cards have a French word? $\newline$ Write your answer as a decimal rounded to the nearest thousandth. $\newline$ ____ $\newline$

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Math Problems

Algebra 2

Find probabilities using the binomial distribution

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

40\%

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

\newline

If Eva picks a study card from the deck

4

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

4

cards have a French word?

\newline

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

\newline

____

\newline

Identify values: Identify the values of $n$ , $k$ , and $p$ for the binomial probability formula. $\newline$ $n = 4$ (number of trials) $\newline$ $k = 4$ (number of successful French word picks) $\newline$ $p = 0.40$ (probability of picking a French card)
Use binomial formula: Use the binomial probability formula $P(X = k) = C(n, k) \cdot (p)^k \cdot (1-p)^{(n-k)}$ . Substitute $n = 4$ , $k = 4$ , and $p = 0.40$ into the formula. $P(X = 4) = C(4, 4) \cdot (0.40)^4 \cdot (1 - 0.40)^{(4 - 4)}$
Calculate $C(4, 4)$ : Calculate the value of $C(4, 4)$ . $C(4, 4) = \frac{4!}{4!(4 - 4)!} = 1$
Solve $(0.40)^4$ : Solve $(0.40)^4$ . $(0.40)^4 = 0.40 \times 0.40 \times 0.40 \times 0.40 = 0.0256$
Simplify $(1 - 0.40)^{(4 - 4)}$ : Simplify $(1 - 0.40)^{(4 - 4)}$ . $(1 - 0.40)^{(4 - 4)} = (0.60)^0 = 1$
Multiply values to find probability: Multiply all the values together to find the probability. $\newline$ $P(X = 4) = 1 \times 0.0256 \times 1 = 0.0256$ $\newline$ Round the answer to the nearest thousandth. $\newline$ $P(X = 4) = 0.026$