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Last year, Ashley 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, $23\%$ of which have a Latin root. $\newline$ If Ashley randomly chooses a word to study from the list $5$ different times this month, what is the probability that exactly $3$ of the words have a Latin root? $\newline$ Write your answer as a decimal rounded to the nearest thousandth. $\newline$ ____

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Find probabilities using the binomial distribution

Full solution

Q. Last year, Ashley 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,

23\%

of which have a Latin root.

\newline

If Ashley randomly chooses a word to study from the list

5

different times this month, what is the probability that exactly

3

of the words have a Latin root?

\newline

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

\newline

____

Calculate Probability: Now, we need to calculate the probability of picking exactly $3$ words with a Latin root out of $5$ tries. 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, $p$ is the probability of success, and $\binom{n}{k}$ is the binomial coefficient.
Calculate Binomial Coefficient: Calculate the binomial coefficient for $5$ choose $3$ . This is $\frac{5!}{3! \times (5-3)!}$ , which is $10$ .
Calculate Probability of Success: Now, calculate the probability of getting exactly $3$ words with a Latin root. Using the binomial probability formula, we get $P(X=3) = \binom{5}{3} \times (0.23^3) \times (0.77^2)$ .
Plug in Numbers: Plug in the numbers: $P(X=3) = 10 \times (0.23^3) \times (0.77^2)$ .
Calculate Powers: Calculate the powers: $(0.23^3) = 0.012167$ and $(0.77^2) = 0.5929$ .
Multiply Everything: Multiply everything together: $P(X=3) = 10 \times 0.012167 \times 0.5929$ .
Perform Multiplication: Perform the multiplication: $P(X=3) = 10 \times 0.012167 \times 0.5929 = 0.0721$ .

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