A box contains 4 red blocks and 6 yellow blocks. A block is selected at random from the bo… — GRE General Test

A box contains $4$ red blocks and $6$ yellow blocks. A block is selected at random from the box, its color is noted, and it is returned to the box. A second block is then selected at random from the box. What is the probability that both selected blocks are red?

$\frac{4}{25}$ Answer
B
$\frac{2}{15}$
C
$\frac{2}{5}$
D
$\frac{4}{9}$
E
$\frac{4}{5}$

Answer

The probability that both selected blocks are red is

\frac{4}{25}

Because the first block is returned to the box before the second selection, the two draws are independent events. The probability of selecting a red block on any single draw is

\frac{4}{10} = \frac{2}{5}

. Applying the multiplication rule for independent events gives

P(\text{Both red}) = \frac{2}{5} \times \frac{2}{5} = \frac{4}{25}

Step-by-Step Solution

Determine the total number of blocks in the box.

The total number of blocks is

4 + 6 = 10

Probability requires finding the ratio of favorable outcomes to total possible outcomes.

Calculate the probability of drawing a red block on the first selection.

P(\text{First is red}) = \frac{4}{10} = \frac{2}{5}

There are

4

red blocks out of

10

total blocks.

Calculate the probability of drawing a red block on the second selection.

Since the first block is returned to the box, the events are independent, so

P(\text{Second is red}) = \frac{4}{10} = \frac{2}{5}

Replacement preserves the original sample space composition.

Apply the multiplication rule for independent events.

P(\text{Both are red}) = P(\text{First is red}) \times P(\text{Second is red}) = \frac{2}{5} \times \frac{2}{5} = \frac{4}{25}

The probability of two independent events both occurring is the product of their individual probabilities.

Key Concept

Probability of Independent Events

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic