An art gallery features a collection of 60 paintings, each categorized by its medium (oil … — ACT

An art gallery features a collection of $60$ paintings, each categorized by its medium (oil or watercolor) and its historical period ( $19\text{th}$ -century or $20\text{th}$ -century). Among the collection, $35$ paintings are $19\text{th}$ -century works, $40$ paintings are oil paintings, and $10$ paintings are $20\text{th}$ -century watercolor works. If a single painting is selected at random from the collection, what is the probability that the selected painting is a $19\text{th}$ -century oil painting?

A
$\frac{1}{6}$
B
$\frac{1}{4}$
C
$\frac{7}{18}$
$\frac{5}{12}$ Cevap
E
$\frac{1}{2}$

Cevap

\frac{5}{12}

To find the probability of choosing a

19\text{th}

-century oil painting, determine the number of paintings that fit both criteria. Out of

60

paintings,

40

are oil, meaning

20

are watercolors. Given that

10

watercolors are

20\text{th}

-century, the remaining

10

watercolors must be

19\text{th}

-century. Since there are

35

total

19\text{th}

-century paintings, subtracting the

10

19\text{th}

-century watercolors yields

25

19\text{th}

-century oil paintings. The probability is therefore

\frac{25}{60}

, which simplifies to

\frac{5}{12}

Adım Adım Çözüm

Find the total number of watercolor paintings in the collection.

Since there are

60

total paintings and

40

oil paintings, the number of watercolor paintings is

60 - 40 = 20

Paintings are categorized into oil or watercolor, so the remaining paintings must be watercolors.

Find the number of

19\text{th}

-century watercolor paintings.

There are

20

total watercolor paintings and

10

are from the

20\text{th}

century, so the number of

19\text{th}

-century watercolors is

20 - 10 = 10

Subtracting the

20\text{th}

-century watercolors from the total watercolors gives the

19\text{th}

-century watercolors.

Calculate the number of

19\text{th}

-century oil paintings.

There are

35

total

19\text{th}

-century paintings. Subtracting the

10

19\text{th}

-century watercolors leaves

35 - 10 = 25

19\text{th}

-century oil paintings.

All

19\text{th}

-century paintings are either oil or watercolor.

Compute the probability of picking a

19\text{th}

-century oil painting.

The probability is

\frac{25}{60} = \frac{5}{12}

Divide the number of favorable outcomes (

25

) by the total number of paintings in the sample space (

60

Anahtar Kavram

Two-way categorization and conditional counting for compound probabilities

Tahmini Süre:2m 0s

Soru

Cevap

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