There are $13$ college students running for student government class president. The candidates include $5$ biology majors. $\newline$ If $9$ of the candidates are randomly chosen to give the first $9$ speeches, what is the probability that exactly $2$ of the chosen candidates are biology majors? $\newline$ Write your answer as a decimal rounded to four decimal places. $\newline$ $\square$

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Find probabilities using combinations and permutations

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Q. There are

13

college students running for student government class president. The candidates include

5

biology majors.

\newline

9

of the candidates are randomly chosen to give the first

9

speeches, what is the probability that exactly

2

of the chosen candidates are biology majors?

\newline

Write your answer as a decimal rounded to four decimal places.

\newline

\square

Identify Total Number: Identify the total number of ways to choose $9$ candidates out of $13$ without regard to the major. $\newline$ We use the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$ where $n$ is the total number of items, $k$ is the number of items to choose, and “!” denotes factorial. $\newline$ Total number of ways to choose $9$ candidates out of $13$ is $C(13, 9)$ . $\newline$ Calculate $C(13, 9) = \frac{13!}{9!(13-9)!} = \frac{13!}{9!4!} = \frac{13\times12\times11\times10}{4\times3\times2\times1}$ .
Calculate Combination: Identify the number of ways to choose $2$ biology majors out of $5$ . We use the same combination formula $C(n, k)$ . Number of ways to choose $2$ biology majors out of $5$ is $C(5, 2)$ . Calculate $C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{(5\times4)}{(2\times1)}$ .
Identify Number Biology Majors: Identify the number of ways to choose the remaining $7$ candidates from the non-biology majors. $\newline$ There are $13 - 5 = 8$ non-biology majors. $\newline$ Number of ways to choose $7$ candidates out of $8$ non-biology majors is $C(8, 7)$ . $\newline$ Calculate $C(8, 7) = \frac{8!}{(7!(8-7)!)} = \frac{8!}{(7!1!)} = \frac{8}{1}.$
Calculate Number Non-Biology Majors: Calculate the number of ways to choose exactly $2$ biology majors and $7$ non-biology majors. $\newline$ Multiply the results from Step $2$ and Step $3$ . $\newline$ Number of favorable outcomes = $C(5, 2) \times C(8, 7)$ . $\newline$ Calculate the number of favorable outcomes = $(5\times4) / (2\times1) \times 8 / 1 = 10 \times 8 = 80$ .
Calculate Number Favorable Outcomes: Calculate the probability that exactly $2$ of the chosen candidates are biology majors. $\newline$ Probability $= \frac{\text{Number of favorable outcomes}}{\text{Total number of ways to choose } 9 \text{ candidates}}$ . $\newline$ Calculate the probability $= \frac{80}{C(13, 9)}$ . $\newline$ From Step $1$ , we have $C(13, 9) = \frac{(13\times12\times11\times10)}{(4\times3\times2\times1)}$ . $\newline$ Calculate the probability $= \frac{80}{\left(\frac{(13\times12\times11\times10)}{(4\times3\times2\times1)}\right)}$ .
Calculate Probability: Simplify the probability and round to four decimal places. $\newline$ Calculate the probability = $\frac{80}{\left(\frac{13\times12\times11\times10}{4\times3\times2\times1}\right)} = \frac{80}{13\times12\times11\times10} \times \frac{4\times3\times2\times1}{1}$ . $\newline$ Simplify the probability = $\frac{80}{17160} = 0.004662004662004662$ . $\newline$ Round to four decimal places = $0.0047$ .