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As Student Council President, Marcy organized a school-wide scavenger hunt. She hid $8$ goody bags, $6$ of which were hidden in the main office. If $5$ goody bags were randomly discovered during first period, what is the probability that all of them were hidden in the main office? $\newline$ Write your answer as a decimal rounded to four decimal places._____

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Find probabilities using combinations and permutations

Full solution

Q. As Student Council President, Marcy organized a school-wide scavenger hunt. She hid

8

goody bags,

6

of which were hidden in the main office. If

5

goody bags were randomly discovered during first period, what is the probability that all of them were hidden in the main office?

\newline

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

Total Goody Bags: Total number of goody bags: $8$ $\newline$ Goody bags to be found: $5$ $\newline$ Calculate total possible outcomes using combinations. $\newline$ Total outcomes: $\binom{8}{5}$
Calculate Total Outcomes: Find the value of $\binom{8}{5}$ . $\binom{8}{5} = \frac{8!}{5!(8-5)!} =\frac{8!}{5!3!} =\frac{8 \times 7 \times 6 \times 5!}{5! \times 3 \times 2 \times 1} = 56$
Favorable Outcomes from Main Office: Number of goody bags in the main office: $6$ $\newline$ Goody bags to be found from the main office: $5$ $\newline$ Calculate favorable outcomes using combinations. $\newline$ Favorable outcomes: ${}_{6}C_{5}$
Calculate Probability: Find the value of $\binom{6}{5}$ . $\newline$ $\binom{6}{5} = \frac{6!}{5!(6-5)!} =\frac{6!}{5!1!} = \frac{6 \times 5!}{5! \times 1} = 6$
Calculate Probability: Find the value of $_6C_5$ . $_6C_5 = \frac{6!}{5!(6-5)!} =\frac{6!}{5!1!} = \frac{6 \times 5!}{5! \times 1} = 6$ Calculate the probability. Probability = Favorable outcomes / Total possible outcomes $= \frac{6}{56} = 0.1071$ when rounded to four decimal places.

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