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After touring a coffee bean farm, Lily impulsively bought $8$ bags of coffee beans. Of the bags she bought, $6$ were filled with caramel flavored coffee beans. $\newline$ If Lily randomly selects $5$ of the bags to send to family members, what is the probability that all of them are caramel flavored? $\newline$ Write your answer as a decimal rounded to four decimal places. $\newline$ ____ $\newline$

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Find probabilities using combinations and permutations

Full solution

Q. After touring a coffee bean farm, Lily impulsively bought

8

bags of coffee beans. Of the bags she bought,

6

were filled with caramel flavored coffee beans.

\newline

If Lily randomly selects

5

of the bags to send to family members, what is the probability that all of them are caramel flavored?

\newline

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

\newline

____

\newline

Calculate Total Ways: Total number of bags: $8$ $\newline$ Bags to choose for sending: $5$ $\newline$ Calculate the total number of ways to choose $5$ bags out of $8$ . $\newline$ Total outcomes: $_8C_5$
Calculate Value: Find the value of ${}_8C_5$ . ${}_8C_5 = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = \frac{8 \times 7 \times 6 \times 5!}{5! \times 3 \times 2 \times 1} = 56$
Choose Caramel Bags: Number of caramel flavored bags: $6$ Number of chosen caramel flavored bags: $5$ Calculate the number of ways to choose $5$ caramel flavored bags out of $6$ . Favorable outcomes: $_{6}C_{5}$
Calculate Value: Find the value of $\binom{6}{5}$ . $\newline$ $\binom{6}{5} = \frac{6!}{5!(6-5)!}$ $\newline$ $= \frac{6!}{5!1!}$ $\newline$ $= \frac{6 \times 5!}{5! \times 1}$ $\newline$ $= 6$
Calculate Probability: Calculate the probability that all selected bags are caramel flavored. $\newline$ Probability = Favorable outcomes / Total possible outcomes $\newline$ = $\frac{6}{56}$ $\newline$ $\approx 0.1071$

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