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An English teacher gave his class a choice of $8$ books to read, $6$ of which are contemporary novels. If a student randomly picks $4$ books, what is the probability that all of them are contemporary novels? 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. An English teacher gave his class a choice of

8

books to read,

6

of which are contemporary novels. If a student randomly picks

4

books, what is the probability that all of them are contemporary novels? Write your answer as a decimal rounded to four decimal places. ____

Total Books and Choices: Total number of books: $8$ Books to be chosen: $4$ Calculate total possible outcomes using combinations. Total outcomes: $_{8}C_{4}$
Calculate Total Outcomes: Find the value of $\binom{8}{4}$ . $\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70$
Calculate Favorable Outcomes: Number of contemporary novels: $6$ Books to be chosen that are contemporary novels: $4$ Calculate favorable outcomes using combinations. Favorable outcomes: $_{6}C_{4}$
Find Probability: Find the value of $_6C_4$ . $_6C_4 = \frac{6!}{4!(6-4)!} = \frac{6!}{4!2!} = \frac{6 \times 5}{2 \times 1} = 15$
Find Probability: Find the value of $\binom{6}{4}$ . $\binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6!}{4!2!} = \frac{6 \times 5}{2 \times 1} = 15$ Calculate the probability that all $4$ books are contemporary novels. $\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total possible outcomes}} = \frac{15}{70} \approx 0.2143$

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