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Lucy and her dad went ice-fishing this winter. They caught $8$ total fish, including $6$ arctic char. If on the last day of the trip, Lucy randomly selected $4$ fish to donate to fishermen who hadn't caught any fish that day, what is the probability that all of them are arctic char? $\newline$ 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. Lucy and her dad went ice-fishing this winter. They caught

8

total fish, including

6

arctic char. If on the last day of the trip, Lucy randomly selected

4

fish to donate to fishermen who hadn't caught any fish that day, what is the probability that all of them are arctic char?

\newline

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

Total Fish Count: Total number of fish: $8$ $Arctic char: \$6$ \)Fish to donate: $4$ \)Calculate total possible combinations of choosing $4$ fish from $8$ ._\(8C_$4$ = \frac{ $8!$ }{ $4!(8-4)!$ }\)
Calculate Total Combinations: ${}_8C_4 = \frac{8!}{4!4!}$ $= \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1}$ $= 70$
Calculate Arctic Char Combinations: Calculate combinations of choosing $4$ arctic char from $6$ .
\substack{6\4}C = \frac{6!}{4!(6-4)!}
Calculate Probability: $\binom{6}{4} = \frac{6!}{4!2!}$ $= \frac{6 \times 5}{2 \times 1}$ $= 15$
Calculate Probability: ${}_6C_4 = \frac{6!}{4!2!}$ $= \frac{6 \times 5}{2 \times 1}$ $= 15$ Probability = Favorable outcomes / Total possible outcomes $= \frac{15}{70}$ $= 0.2143$ when rounded to four decimal places.

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