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Kelly teaches a ceramics class on the weekends. In Sunday's class, the students created $8$ clay objects, $6$ of which were plates. After each class, Kelly bakes the clay objects in a hot kiln to harden them. Students pick up their finished pieces later in the week. $\newline$ If Kelly randomly chose $5$ clay objects to bake in the kiln Sunday night, what is the probability that all of them are plates? $\newline$ Write your answer as a decimal rounded to four decimal places. $\newline$ ____ $\newline$

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Find probabilities using combinations and permutations

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Q. Kelly teaches a ceramics class on the weekends. In Sunday's class, the students created

8

clay objects,

6

of which were plates. After each class, Kelly bakes the clay objects in a hot kiln to harden them. Students pick up their finished pieces later in the week.

\newline

If Kelly randomly chose

5

clay objects to bake in the kiln Sunday night, what is the probability that all of them are plates?

\newline

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

\newline

____

\newline

Total Clay Objects: Total number of clay objects: $8$ $Plates among them: \$6$ \)Objects to be chosen for baking: $5$ \)Calculate the total number of ways to choose $5$ objects from $8$ .\)Total outcomes: $_{8}C_{5}$
Calculate Total Ways: Find the value of ${}_8C_5$ . ${}_8C_5 = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = \frac{8 \times 7 \times 6 \times 5!}{5! \times 3 \times 2 \times 1} = 56$
Calculate Value: Calculate the number of ways to choose $5$ plates from $6$ . $\newline$ Favorable outcomes: $\binom{6}{5}$
Choose Plates: Find the value of ${}_6C_5$ . $\newline$ ${}_6C_5 = \frac{6!}{5!(6-5)!} = \frac{6!}{5!1!} = \frac{6 \times 5!}{5! \times 1} = 6$
Calculate Probability: Calculate the probability that all chosen objects are plates. $\newline$ Probability = $\frac{\text{Favorable outcomes}}{\text{Total outcomes}}$ $\newline$ = $\frac{6}{56}$ $\newline$ = $0.1071$ when rounded to four decimal places

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