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Every attendant at a town's Chili Cook-off received a raffle ticket. There were $8$ raffle prizes, including $6$ that were gift certificates to restaurants. $\newline$ If $4$ prizes were randomly raffled away in the first hour of the cook-off, what is the probability that all of them are gift certificates to restaurants? $\newline$ 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. Every attendant at a town's Chili Cook-off received a raffle ticket. There were

8

raffle prizes, including

6

that were gift certificates to restaurants.

\newline

If

4

prizes were randomly raffled away in the first hour of the cook-off, what is the probability that all of them are gift certificates to restaurants?

\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: $4$ Calculate the total number of ways to choose $4$ prizes out of $8$ . Use the combination formula: $\binom{8}{4}$ .
Calculate ${}_8C_4$ : Find the value of ${}_8C_4$ .
{}_8C_4 = \frac{\(8\)!}{\(4\)!(\(8\)\(-4\))!} = \frac{\(8\) \times \(7\) \times \(6\) \times \(5\)}{\(4\) \times \(3\) \times \(2\) \times \(1\)} = \(70
Gift Certificates Calculation: Number of gift certificates: $6$ Number of gift certificates to be raffled: $4$ Calculate the number of ways to choose $4$ gift certificates out of $6$ . Use the combination formula: $\binom{6}{4}$ .
Calculate _$6$C_$4$: Find the value of _$6$C_$4$. $\newline$ $\binom{6}{4} = \frac{6!}{4!(6-4)!}$ $\newline$ = $\frac{6 \times 5}{2 \times 1}$ $\newline$ = $15$
Calculate Probability: Calculate the probability that all $4$ raffled prizes are gift certificates. $\newline$ Probability = $\frac{\text{Favorable outcomes}}{\text{Total possible outcomes}}$ $\newline$ = $\frac{15}{70}$ $\newline$ = $0.2143$

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