A jar contains 3 red marbles and 7 blue marbles. A marble is drawn at random from the jar,… — GRE General Test

A jar contains $3$ red marbles and $7$ blue marbles. A marble is drawn at random from the jar, its color is noted, and it is returned to the jar. A second marble is then drawn at random. What is the probability that both drawn marbles are red?

$\frac{9}{100}$ Cevap
B
$\frac{3}{10}$
C
$\frac{3}{5}$
D
$\frac{9}{49}$
E
$\frac{6}{7}$

Cevap

The probability that both drawn marbles are red is

\frac{9}{100}

The option stating

\frac{9}{100}

is correct because the two draws are independent due to replacement. The probability of getting a red marble on any single draw is

\frac{3}{10}

. By the multiplication rule for independent events, the probability of both events occurring is

\frac{3}{10} \times \frac{3}{10} = \frac{9}{100}

Adım Adım Çözüm

Calculate the total number of marbles in the jar.

3 \text{ red} + 7 \text{ blue} = 10 \text{ total marbles}

Probability requires knowing the size of the full sample space.

Find the probability of drawing a red marble on a single draw.

P(\text{Red}) = \frac{3}{10}

There are

3

favorable outcomes (red marbles) out of

10

total possible outcomes.

Apply the multiplication rule for independent events.

P(\text{Red}_1 \text{ and } \text{Red}_2) = \frac{3}{10} \times \frac{3}{10} = \frac{9}{100}

Because the first marble is replaced, the second draw is independent of the first, so joint probability is the product of individual probabilities.

Anahtar Kavram

Multiplication Rule for Independent Events:

P(A \text{ and } B) = P(A) \times P(B)

when events

A

and

B

are independent.

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