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Cora organized a technology career fair at her high school. She did an excellent job recruiting $8$ guest speakers, $5$ of whom worked in construction technology. $\newline$ If Cora randomly assigns $4$ speakers to speak during the afternoon session, what is the probability that all of them work in construction technology? $\newline$ Write your answer as a decimal rounded to four decimal places. $\newline$ ____ $\newline$

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Find probabilities using combinations and permutations

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Q. Cora organized a technology career fair at her high school. She did an excellent job recruiting

8

guest speakers,

5

of whom worked in construction technology.

\newline

If Cora randomly assigns

4

speakers to speak during the afternoon session, what is the probability that all of them work in construction technology?

\newline

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

\newline

____

\newline

Calculate Total Ways: First, calculate the total number of ways to choose $4$ speakers out of $8$ without regard to order. This is a combination problem, so use the formula for combinations: $C(n, k) = \frac{n!}{k!(n-k)!}$ , where $n$ is the total number of items, and $k$ is the number of items to choose. $\newline$ $C(8, 4) = \frac{8!}{4!(8-4)!} = \frac{(8\times7\times6\times5)}{(4\times3\times2\times1)} = 70$ .
Choose Speakers from Construction: Next, calculate the number of ways to choose $4$ speakers from the $5$ who work in construction technology. Again, use the combination formula. $\newline$ $C(5, 4) = \frac{5!}{4!(5-4)!} = \frac{(5)}{(1)} = 5$ .
Calculate Probability: Now, find the probability by dividing the number of successful outcomes by the total number of possible outcomes. $\text{Probability} = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}} = \frac{C(5, 4)}{C(8, 4)}.$ $\text{Probability} = \frac{5}{70}.$
Convert to Decimal: Finally, convert the fraction to a decimal and round to four decimal places. $\newline$ Probability = $\frac{5}{70} \approx 0.0714$ .

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