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As Student Council President, Mona organized a school-wide scavenger hunt. She hid $8$ goody bags, $5$ of which were hidden in the computer lab. If $4$ goody bags were randomly discovered during first period, what is the probability that all of them were hidden in the computer lab? $\newline$ 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. As Student Council President, Mona organized a school-wide scavenger hunt. She hid

8

goody bags,

5

of which were hidden in the computer lab. If

4

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

\newline

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

Calculate Probability: First, calculate the probability of picking $1$ goody bag from the computer lab. $\newline$ There are $5$ goody bags in the computer lab out of $8$ total goody bags. $\newline$ So, the probability for the first goody bag is $\frac{5}{8}$ .
First Goody Bag: Next, since one goody bag has been taken, there are now $4$ goody bags in the computer lab and $7$ total. $\newline$ The probability for the second goody bag is then $\frac{4}{7}$ .
Second Goody Bag: For the third goody bag, now there are $3$ in the computer lab and $6$ total. $\newline$ The probability for the third goody bag is $\frac{3}{6}$ .
Third Goody Bag: Finally, for the fourth goody bag, there are $2$ in the computer lab and $5$ total. $\newline$ The probability for the fourth goody bag is $\frac{2}{5}$ .
Fourth Goody Bag: Now, multiply all the probabilities together to get the total probability. $\newline$ $(\frac{5}{8}) \times (\frac{4}{7}) \times (\frac{3}{6}) \times (\frac{2}{5}) = (\frac{5 \times 4 \times 3 \times 2}{8 \times 7 \times 6 \times 5})$
Multiply Probabilities: Simplify the multiplication and division. $\newline$ $(5 \times 4 \times 3 \times 2) / (8 \times 7 \times 6 \times 5) = (120) / (1680)$
Simplify Multiplication: Now, divide $120$ by $1680$ to get the decimal. $\newline$ $120 / 1680 = 0.07142857142857142$
Divide to Get Decimal: Round the decimal to four decimal places. $0.07142857142857142$ rounded to four decimal places is $0.0714$ .

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