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Last year, Melissa came in $2$ nd place in her school's Spelling Bee. This year, she plans to win $1$ st place. She compiled a long list of words to study, $49\%$ of which have a Latin root. $\newline$ If Melissa randomly chooses a word to study from the list $5$ different times this month, what is the probability that exactly $2$ of the words have a Latin root? $\newline$ 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. Last year, Melissa came in

2

nd place in her school's Spelling Bee. This year, she plans to win

1

st place. She compiled a long list of words to study,

49\%

of which have a Latin root.

\newline

If Melissa randomly chooses a word to study from the list

5

different times this month, what is the probability that exactly

2

of the words have a Latin root?

\newline

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

\newline

____

\newline

Find Probability Latin Root: First, we need to find the probability of choosing a word with a Latin root. Since $49\%$ of the words have a Latin root, the probability is $0.49$ for each pick.
Calculate Not Latin Root: Now, we calculate the probability of not choosing a word with a Latin root, which is $1 - 0.49 = 0.51$ .
Binomial Probability Formula: We're looking for the probability of exactly $2$ words having a Latin root out of $5$ picks. This is a binomial probability problem, where we use the formula $P(X=k) = \binom{n}{k} \cdot (p^k) \cdot ((1-p)^{(n-k)})$ , where $n$ is the number of trials, $k$ is the number of successes, and $p$ is the probability of success.
Calculate $5$ Choose $2$ : Let's calculate ＂ $5$ choose $2$ ＂ which is the number of ways to choose $2$ successes out of $5$ trials. This is $\frac{5!}{2! \times (5-2)!} = 10$ .
Calculate Probability $2$ Latin Root: Now, we calculate the probability of getting exactly $2$ words with a Latin root. Using the binomial formula: $P(X=2) = \binom{5}{2} \times (0.49^2) \times (0.51^3)$ .
Calculate Probability $2$ Latin Root: Now, we calculate the probability of getting exactly $2$ words with a Latin root. Using the binomial formula: $P(X=2) = \binom{5}{2} \times (0.49^2) \times (0.51^3)$ . Plugging in the numbers: $P(X=2) = 10 \times (0.49^2) \times (0.51^3)$ . Let's do the math: $P(X=2) = 10 \times (0.2401) \times (0.132651)$ .

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