A bookstore received a shipment of 50 new books. Among these books, 24 are fiction, 20 are… — ACT

A bookstore received a shipment of $50$ new books. Among these books, $24$ are fiction, $20$ are hardcover, and $8$ are hardcover fiction books. If $1$ book is selected at random from the shipment, what is the probability that the selected book is either a fiction book or a hardcover book?

A
$\frac{4}{25}$
B
$\frac{14}{25}$
$\frac{18}{25}$ Cevap
D
$\frac{22}{25}$
E
$\frac{9}{11}$

Cevap

\frac{18}{25}

To find the probability that a randomly chosen book is either fiction or hardcover, calculate the total number of books satisfying at least one condition. By the Principle of Inclusion-Exclusion, the number of favorable books is

24 + 20 - 8 = 36

. Dividing this by the total of

50

books yields

\frac{36}{50}

, which simplifies to

\frac{18}{25}

Adım Adım Çözüm

Identify the total number of items in the sample space and the counts for each event.

Total books

N = 50

, Fiction

N(F) = 24

, Hardcover

N(H) = 20

, Both

N(F \cap H) = 8

The sample space consists of all 50 shipment books.

Apply the Inclusion-Exclusion Principle to find the number of favorable outcomes.

N(F \cup H) = N(F) + N(H) - N(F \cap H) = 24 + 20 - 8 = 36

Hardcover fiction books are counted in both individual totals, so their intersection must be subtracted once to avoid double-counting.

Calculate the probability and simplify the fraction.

P(F \cup H) = \frac{36}{50} = \frac{18}{25}

Probability is the ratio of favorable outcomes to total outcomes.

Anahtar Kavram

Probability of non-mutually exclusive events using the Addition Rule:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

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