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During her semester living abroad, Cindy made a list of famous buildings to visit. Of the $7$ buildings on her list, $5$ were examples of Gothic architecture. $\newline$ If Cindy randomly chose $4$ buildings to visit during the first half of the semester, what is the probability that all of them are examples of Gothic architecture? $\newline$ 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. During her semester living abroad, Cindy made a list of famous buildings to visit. Of the

7

buildings on her list,

5

were examples of Gothic architecture.

\newline

If Cindy randomly chose

4

buildings to visit during the first half of the semester, what is the probability that all of them are examples of Gothic architecture?

\newline

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

\newline

____

\newline

Calculate Probability: First, calculate the probability of choosing a Gothic building on the first try. $\newline$ There are $5$ Gothic buildings out of $7$ total, so the probability is $\frac{5}{7}$ .
Second Choice Probability: Next, if one Gothic building has been chosen, there are now $4$ Gothic buildings left out of $6$ total buildings. $\newline$ The probability for the second choice is then $\frac{4}{6}$ .
Third Choice Probability: For the third choice, there are $3$ Gothic buildings left out of $5$ total buildings. $\newline$ The probability for the third choice is $\frac{3}{5}$ .
Fourth Choice Probability: Finally, for the fourth choice, there are $2$ Gothic buildings left out of $4$ total buildings. $\newline$ The probability for the fourth choice is $\frac{2}{4}$ .
Multiply Probabilities: Multiply all the probabilities together to get the overall probability. $\newline$ $(\frac{5}{7}) \times (\frac{4}{6}) \times (\frac{3}{5}) \times (\frac{2}{4}) = (\frac{5 \times 4 \times 3 \times 2}{7 \times 6 \times 5 \times 4})$
Simplify Multiplication: Simplify the multiplication and division. $\newline$ The $5$ s and the $4$ s cancel out, so we're left with $(3 \times 2) / (7 \times 6)$ .
Calculate Final Probability: Calculate the final probability. $\newline$ $(3 \times 2) / (7 \times 6) = 6 / 42$
Simplify Fraction: Simplify the fraction $\frac{6}{42}$ to its lowest terms. $\newline$ $\frac{6}{42} = \frac{1}{7}$

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