Adriana and her best friend wanted to paint their apartment. In their storage closet, they found $9$ cans of paint. Adriana remembered that $7$ of the cans contained light blue paint, but none of the cans had labels. $\newline$ If Adriana randomly opened $6$ cans, what is the probability that all of them contain light blue paint? $\newline$ 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. Adriana and her best friend wanted to paint their apartment. In their storage closet, they found

9

cans of paint. Adriana remembered that

7

of the cans contained light blue paint, but none of the cans had labels.

\newline

If Adriana randomly opened

6

cans, what is the probability that all of them contain light blue paint?

\newline

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

\newline

____

\newline

Calculate probability of first can: Calculate the probability of the first can being light blue. Since there are $7$ light blue cans out of $9$ total cans, the probability is $\frac{7}{9}$ .
Calculate probability of second can: Calculate the probability of the second can being light blue after one light blue can is already taken out. $\newline$ Now there are $6$ light blue cans left and $8$ cans in total, so the probability is $\frac{6}{8}$ .
Calculate probability of third can: Calculate the probability of the third can being light blue. Now there are $5$ light blue cans left and $7$ cans in total, so the probability is $\frac{5}{7}$ .
Calculate probability of fourth can: Calculate the probability of the fourth can being light blue. Now there are $4$ light blue cans left and $6$ cans in total, so the probability is $\frac{4}{6}.$
Calculate probability of fifth can: Calculate the probability of the fifth can being light blue. Now there are $3$ light blue cans left and $5$ cans in total, so the probability is $\frac{3}{5}$ .
Calculate probability of sixth can: Calculate the probability of the sixth can being light blue. Now there are $2$ light blue cans left and $4$ cans in total, so the probability is $\frac{2}{4}$ .
Multiply all probabilities: Multiply all the probabilities together to get the overall probability. $\newline$ $(\frac{7}{9}) \times (\frac{6}{8}) \times (\frac{5}{7}) \times (\frac{4}{6}) \times (\frac{3}{5}) \times (\frac{2}{4}) = \frac{7}{9} \times \frac{3}{4} \times 1 \times \frac{2}{3} \times \frac{3}{5} \times \frac{1}{2}$
Simplify multiplication of fractions: Simplify the multiplication of the fractions. $\newline$ $\frac{7}{9} \times \frac{3}{4} \times \frac{2}{3} \times \frac{3}{5} \times \frac{1}{2} = \frac{7 \times 3 \times 2 \times 3 \times 1}{9 \times 4 \times 3 \times 5 \times 2}$ $\newline$ $= \frac{7 \times 1 \times 1 \times 1}{3 \times 2 \times 5 \times 2}$ $\newline$ $= \frac{7}{60}$
Convert fraction to decimal: Convert the fraction to a decimal and round to four decimal places. $\newline$ $\frac{7}{60} = 0.1167$