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Difficulty: HardBasic Probability and Counting Methods

A community library's summer program requires participants to create a 3-book reading list consisting of exactly 11 biography, 11 science fiction novel, and 11 history book selected from a featured list. The featured list contains 55 biographies (2 of which have over 400400 pages), 66 science fiction novels (3 of which have over 400400 pages), and 44 history books (1 of which has over 400400 pages). If a participant selects 11 book of each genre at random from the featured list, what is the probability that at least 11 of the selected books has over 400400 pages? Express your answer as a decimal.

Answer: 0.775

Answer

0.775
To find the probability of selecting at least one book over 400400 pages, it is most efficient to use the complement rule: P(at least one)=1−P(none)P(\text{at least one}) = 1 - P(\text{none}). There are 5×6×4=1205 \times 6 \times 4 = 120 total 3-book combinations. The number of books with 400400 pages or fewer in each category are 33 biographies, 33 sci-fi novels, and 33 history books. Thus, there are 3×3×3=273 \times 3 \times 3 = 27 combinations with no books over 400400 pages. The probability of choosing no books over 400400 pages is 27120=0.225\frac{27}{120} = 0.225. Subtracting this from 11 gives 1−0.225=0.7751 - 0.225 = 0.775.

Step-by-Step Solution

1
Calculate the total number of possible combinations of selecting 1 book from each genre
Total combinations = 5×6×4=1205 \times 6 \times 4 = 120
By the Fundamental Counting Principle, multiplying the number of choices in each independent category yields the total outcomes.
2
Determine the complement event: selecting a reading list where NO book has over 400 pages
Available books of 400 pages or fewer: 3 biographies, 3 sci-fi novels, and 3 history books
Subtracting the number of books over 400 pages from the total in each category gives the count of books with 400 pages or fewer.
3
Calculate the number of combinations consisting entirely of books with 400 pages or fewer
Complement combinations = 3×3×3=273 \times 3 \times 3 = 27
Applying the Fundamental Counting Principle to the non-qualifying choices gives the total outcomes for the complement event.
4
Find the probability of the complement event and subtract from 1 to find the target probability
P(at least one over 400)=1−27120=1−0.225=0.775P(\text{at least one over } 400) = 1 - \frac{27}{120} = 1 - 0.225 = 0.775
The complement rule states that P(A)=1−P(A′)P(A) = 1 - P(A'), which is much more efficient than calculating probabilities for 1, 2, or 3 long books separately.

Key Concept

Complementary Probability and Fundamental Counting Principle
Estimated Time:1m 30s
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Topics

Basic Probability and Counting Methods

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