A jar contains 4 red marbles, 6 blue marbles, and 5 green marbles. If two marbles are sele… — GRE General Test

A jar contains $4$ red marbles, $6$ blue marbles, and $5$ green marbles. If two marbles are selected at random one after another without replacement, what is the probability that the first marble selected is red and the second marble selected is blue?

$\frac{4}{35}$ Answer
B
$\frac{8}{75}$
C
$\frac{73}{105}$
D
$\frac{12}{55}$
E
$\frac{10}{29}$

Answer

The probability that the first marble selected is red and the second marble selected is blue is

\frac{4}{35}

To find the joint probability of two dependent sequential events, multiply the probability of the first event by the conditional probability of the second event. The probability of drawing a red marble first is

\frac{4}{15}

. Because the selection is made without replacement, there are

14

marbles remaining in the jar,

6

of which are blue. The probability of drawing a blue marble second is

\frac{6}{14}

. Multiplying these together yields

\frac{4}{15} \times \frac{6}{14} = \frac{24}{210} = \frac{4}{35}

Step-by-Step Solution

Calculate the total number of marbles in the jar initially.

Total marbles =

4 + 6 + 5 = 15

The sample space size for the first draw is the sum of all marbles.

Determine the probability of selecting a red marble on the first draw.

P(\text{Red}_1) = \frac{4}{15}

There are

4

favorable outcomes out of

15

total outcomes.

Determine the probability of selecting a blue marble on the second draw given that one red marble was removed without replacement.

P(\text{Blue}_2 \mid \text{Red}_1) = \frac{6}{14} = \frac{3}{7}

After removing one red marble, there are still

6

blue marbles, but only

14

total marbles remaining.

Apply the multiplication rule for dependent events.

P(\text{Red}_1 \text{ and } \text{Blue}_2) = \frac{4}{15} \times \frac{6}{14} = \frac{24}{210} = \frac{4}{35}

The joint probability of sequential dependent events is the product of the first event's probability and the conditional probability of the second event.

Key Concept

Conditional Probability and Dependent Events

Estimated Time:1m 30s

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