A container holds 40 marbles, each of which is red, blue, or green. The probability of ran… — ACT

A container holds $40$ marbles, each of which is red, blue, or green. The probability of randomly drawing a red marble from the container is $\frac{2}{5}$ . Among the remaining marbles, the ratio of blue marbles to green marbles is $1:3$ . If one marble is selected at random from the container, what is the probability that it is green?

A
$\frac{3}{20}$
B
$\frac{1}{5}$
$\frac{9}{20}$ Answer
D
$\frac{3}{5}$
E
$\frac{3}{4}$

Answer

The probability that the randomly selected marble is green is

\frac{9}{20}

The correct answer is

\frac{9}{20}

. The probability of drawing a non-red marble is

1 - \frac{2}{5} = \frac{3}{5}

. Given that the non-red marbles are split in a

1:3

ratio between blue and green, green marbles represent

\frac{3}{1+3} = \frac{3}{4}

of the non-red group. Multiplying the probability of drawing a non-red marble by the proportion of green marbles within that group gives

\frac{3}{5} \times \frac{3}{4} = \frac{9}{20}

Step-by-Step Solution

Find the total number of red marbles and the number of remaining marbles.

Since the probability of drawing a red marble is

\frac{2}{5}

, the number of red marbles is

\frac{2}{5} \times 40 = 16

. The remaining number of marbles (blue and green) is

40 - 16 = 24

Determining the count of non-red marbles isolates the sample space for blue and green marbles.

Use the ratio of blue to green marbles to find the number of green marbles.

The ratio of blue to green is

1:3

, meaning there are

1 + 3 = 4

equal parts. Each part contains

\frac{24}{4} = 6

marbles. Therefore, there are

3 \times 6 = 18

green marbles.

Converting a part-to-part ratio into actual counts allows calculation of the favorable outcome.

Calculate the probability of randomly drawing a green marble.

The probability is the number of green marbles divided by the total number of marbles:

\frac{18}{40} = \frac{9}{20}

Basic probability is defined as the number of favorable outcomes divided by the total number of possible outcomes.

Key Concept

Combining complement probability rules with part-to-whole ratio conversions to compute event probabilities.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic