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Scott got a summer job at a movie theater cleaning the aisles after each film. This Sunday, $8$ movies are scheduled to show, $5$ of which feature a teenager as the one main protagonist. If Scott is randomly assigned to clean up after $4$ movies, what is the probability that all of them feature a teenager as the one main protagonist? Write your answer as a decimal rounded to four decimal places.._________

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Find probabilities using combinations and permutations

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Q. Scott got a summer job at a movie theater cleaning the aisles after each film. This Sunday,

8

movies are scheduled to show,

5

of which feature a teenager as the one main protagonist. If Scott is randomly assigned to clean up after

4

movies, what is the probability that all of them feature a teenager as the one main protagonist? Write your answer as a decimal rounded to four decimal places.._________

Calculate Total Ways: Calculate the total number of ways to choose $4$ movies out of $8$ . This is a combination problem, so we use the combination formula: $C(n, k) = \frac{n!}{k!(n-k)!}$ $C(8, 4) = \frac{8!}{4!(8-4)!} = \frac{(8\times7\times6\times5)}{(4\times3\times2\times1)} = 70$ ways.
Choose Teen Protagonist: Calculate the number of ways to choose $4$ movies that all feature a teenager as the main protagonist out of the $5$ available. $\newline$ $C(5, 4) = \frac{5!}{4!(5-4)!} = \frac{(5)}{(1)} = 5$ ways.
Find Probability: Find the probability by dividing the number of favorable outcomes by the total number of possible outcomes. $\newline$ Probability = Favorable outcomes / Total outcomes = $\frac{C(5, 4)}{C(8, 4)}$ $\newline$ Probability = $\frac{5}{70}$
Simplify and Round: Simplify the fraction to get the decimal form. $\newline$ Probability = $\frac{5}{70} = 0.07142857...$ $\newline$ Round to four decimal places. $\newline$ Probability $\approx 0.0714$

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