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Sanjay is helping his younger brother wrap presents for the holidays. They have $8$ different rolls of wrapping paper, including $6$ rolls with a comic strip design. $\newline$ If Sanjay's brother randomly chooses $5$ rolls of wrapping paper to wrap the first group of presents, what is the probability that all of them have a comic strip design? $\newline$ Write your answer as a decimal rounded to four decimal places. $\newline$ ____

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Find probabilities using combinations and permutations

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Q. Sanjay is helping his younger brother wrap presents for the holidays. They have

8

different rolls of wrapping paper, including

6

rolls with a comic strip design.

\newline

If Sanjay's brother randomly chooses

5

rolls of wrapping paper to wrap the first group of presents, what is the probability that all of them have a comic strip design?

\newline

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

\newline

____

Total Rolls and Comic Strips: There are $8$ rolls in total and $6$ of them have a comic strip design. We need to calculate the probability of choosing $5$ comic strip designs in a row.
Calculate Probability of One Design: First, find the probability of choosing one comic strip design: $\frac{6}{8}.$
Calculate Probability for Each Roll: Then, since one comic strip roll is taken, there are now $5$ comic strip rolls left out of $7$ total rolls. The probability for the second roll is $\frac{5}{7}$ .
Multiply All Probabilities: Continue this pattern for all $5$ rolls. The third roll's probability is $\frac{4}{6}$ , the fourth is $\frac{3}{5}$ , and the fifth is $\frac{2}{4}$ .
Simplify and Multiply Fractions: Multiply all these probabilities together to get the total probability: $(\frac{6}{8}) \times (\frac{5}{7}) \times (\frac{4}{6}) \times (\frac{3}{5}) \times (\frac{2}{4})$ .
Cancel Common Terms: Simplify the fractions and multiply: $(\frac{3}{4}) \times (\frac{5}{7}) \times (\frac{2}{3}) \times (\frac{3}{5}) \times (\frac{1}{2})$ .
Final Probability Calculation: Cancel out the common terms: $(\frac{3}{4}) \times (\frac{1}{7}) \times (\frac{1}{1}) \times (\frac{1}{1}) \times (\frac{1}{2})$ .
Convert to Decimal: Now multiply the remaining fractions: $(\frac{3}{4}) \times (\frac{1}{7}) \times (\frac{1}{2}) = \frac{3}{56}$ .
Convert to Decimal: Now multiply the remaining fractions: $(\frac{3}{4}) \times (\frac{1}{7}) \times (\frac{1}{2}) = \frac{3}{56}$ . Convert the fraction to a decimal and round to four decimal places: $\frac{3}{56} \approx 0.0536$ .

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