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Joey volunteers at a children's hospital as an Art Director. He brought $8$ T-shirts for the children to decorate this week. $5$ of the T-shirts are a size small. If Joey was in a rush to get to the hospital, and he randomly grabbed $4$ T-shirts to decorate, what is the probability that all of them are small? 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. Joey volunteers at a children's hospital as an Art Director. He brought

8

T-shirts for the children to decorate this week.

5

of the T-shirts are a size small. If Joey was in a rush to get to the hospital, and he randomly grabbed

4

T-shirts to decorate, what is the probability that all of them are small? Write your answer as a decimal rounded to four decimal places.._________

Calculate Probability First T-shirt: First, calculate the probability of picking a small T-shirt on the first try. $\newline$ There are $5$ small T-shirts out of $8$ total T-shirts. $\newline$ Probability for the first T-shirt = Number of small T-shirts / Total number of T-shirts = $\frac{5}{8}$ .
Calculate Probability Second T-shirt: Next, calculate the probability of picking a small T-shirt on the second try, given that the first T-shirt was small. Now there are $4$ small T-shirts left and $7$ T-shirts in total. Probability for the second T-shirt = $\frac{4}{7}$ .
Calculate Probability Third T-shirt: Then, calculate the probability of picking a small T-shirt on the third try, given that the first two T-shirts were small. $\newline$ Now there are $3$ small T-shirts left and $6$ T-shirts in total. $\newline$ Probability for the third T-shirt = $\frac{3}{6}$ .
Calculate Probability Fourth T-shirt: Finally, calculate the probability of picking a small T-shirt on the fourth try, given that the first three T-shirts were small. $\newline$ Now there are $2$ small T-shirts left and $5$ T-shirts in total. $\newline$ Probability for the fourth T-shirt = $\frac{2}{5}$ .
Multiply Probabilities: Multiply all the probabilities together to get the overall probability that all $4$ T-shirts are small. $\newline$ Overall probability = $(\frac{5}{8}) \times (\frac{4}{7}) \times (\frac{3}{6}) \times (\frac{2}{5})$ .
Perform Calculation: Now, perform the calculation. $\newline$ Overall probability = $(\frac{5}{8}) \times (\frac{4}{7}) \times (\frac{1}{2}) \times (\frac{2}{5}) = (\frac{5 \times 4 \times 1 \times 2}{8 \times 7 \times 2 \times 5}) = \frac{40}{560}$ .
Simplify Fraction: Simplify the fraction $\frac{40}{560}$ to its lowest terms. $\newline$ $\frac{40}{560} = \frac{1}{14}.$
Convert to Decimal: Convert the fraction to a decimal and round to four decimal places. $\newline$ $\frac{1}{14}$ as a decimal is approximately $0.0714$ .

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