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Malia is shopping for a costume at a thrift store. There are $8$ costumes hanging on the rack, including $6$ witch costumes. $\newline$ If the costumes all look like the right size, and Malia randomly selects $4$ to try on, what is the probability that all of them are witch costumes? $\newline$ 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. Malia is shopping for a costume at a thrift store. There are

8

costumes hanging on the rack, including

6

witch costumes.

\newline

If the costumes all look like the right size, and Malia randomly selects

4

to try on, what is the probability that all of them are witch costumes?

\newline

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

\newline

____

\newline

Calculate Probability of First Pick: First, calculate the probability of picking a witch costume on the first try. $\newline$ There are $6$ witch costumes out of $8$ , so the probability is $\frac{6}{8}$ .
Calculate Probability of Second Pick: Next, calculate the probability of picking a witch costume on the second try, given that the first one was a witch costume. $\newline$ Now there are $5$ witch costumes left and $7$ costumes in total, so the probability is $\frac{5}{7}$ .
Calculate Probability of Third Pick: Then, calculate the probability of picking a witch costume on the third try, given that the first two were witch costumes. $\newline$ Now there are $4$ witch costumes left and $6$ costumes in total, so the probability is $\frac{4}{6}.$
Calculate Probability of Fourth Pick: Finally, calculate the probability of picking a witch costume on the fourth try, given that the first three were witch costumes. $\newline$ Now there are $3$ witch costumes left and $5$ costumes in total, so the probability is $\frac{3}{5}$ .
Multiply Probabilities: Now, multiply all the probabilities together to get the overall probability of picking $4$ witch costumes in a row. $\frac{6}{8} \times \frac{5}{7} \times \frac{4}{6} \times \frac{3}{5} = \frac{6\times5\times4\times3}{8\times7\times6\times5}$
Simplify Multiplication: Simplify the multiplication and cancel out common factors. $\newline$ The $6s$ and the $5s$ cancel out, leaving $(4 \times 3)/(8 \times 7)$ .
Perform Multiplication: Now, perform the multiplication. $4\times3 = 12$ and $8\times7 = 56$ , so the probability is $\frac{12}{56}.$
Simplify Fraction: Simplify the fraction $\frac{12}{56}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $4$ . $12 \div 4 = 3$ and $56 \div 4 = 14$ , so the simplified probability is $\frac{3}{14}$ .
Convert to Decimal: Finally, convert the fraction to a decimal and round to four decimal places. $\frac{3}{14} \approx 0.2143$

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