A container holds 4 red tokens and 6 blue tokens. If two tokens are selected at random one… — GMAT

A container holds $4$ red tokens and $6$ blue tokens. If two tokens are selected at random one after another without replacement, what is the probability that both selected tokens are red?

A
$\frac{4}{25}$
B
$\frac{8}{45}$
$\frac{2}{15}$ Answer
D
$\frac{2}{5}$
E
$\frac{1}{3}$

Answer

\frac{2}{15}

The option stating

\frac{2}{15}

is correct because selecting without replacement makes the second event dependent on the first. The first probability is

\frac{4}{10}

and the second probability is

\frac{3}{9}

. Multiplying these gives

\frac{4}{10} \times \frac{3}{9} = \frac{12}{90} = \frac{2}{15}

Step-by-Step Solution

Calculate the probability of selecting a red token on the first draw.

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

There are

4

red tokens out of a total of

10

tokens.

Calculate the conditional probability of selecting a red token on the second draw given the first was red.

P(\text{Second Red} \mid \text{First Red}) = \frac{3}{9} = \frac{1}{3}

Because the selection is without replacement,

1

red token and

1

total token have been removed, leaving

3

red tokens out of

9

remaining tokens.

Multiply the sequential probabilities to find the combined probability.

P(\text{Both Red}) = \frac{2}{5} \times \frac{1}{3} = \frac{2}{15}

The probability of both dependent events occurring sequentially is the product of their individual conditional probabilities.

Key Concept

Probability of Dependent Events Without Replacement

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