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Every attendant at a town's Chili Cook-off received a raffle ticket. There were $8$ raffle prizes, including $5$ that were cooking utensils. If $4$ prizes were randomly raffled away in the first hour of the cook-off, what is the probability that all of them are cooking utensils? $\newline$ Write your answer as a decimal rounded to four decimal places.._________

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Find probabilities using combinations and permutations

Full solution

Q. Every attendant at a town's Chili Cook-off received a raffle ticket. There were

8

raffle prizes, including

5

that were cooking utensils. If

4

prizes were randomly raffled away in the first hour of the cook-off, what is the probability that all of them are cooking utensils?

\newline

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

Total Prizes Calculation: Total number of prizes: $8$ Prizes to be raffled in the first hour: $4$ Total outcomes: $\binom{8}{4}$
Calculate Total Outcomes: Calculate $\binom{8}{4}$ : $\newline$ $\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70$
Cooking Utensils Raffle: Number of cooking utensils: $5$ $\newline$ Number of cooking utensils to be raffled: $4$ $\newline$ Favorable outcomes: ${}_{5}C_{4}$
Calculate Favorable Outcomes: Calculate $\binom{5}{4}$ : $\newline$ $\binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5 \times 4!}{4! \times 1} = 5$
Calculate Probability: Probability = $\frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{\(5\)}{\(70\)} = \(0\).\(0714\)

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