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
$\frac{8}{45}$
$\frac{2}{15}$ Cevap
D
$\frac{2}{5}$
E
$\frac{1}{3}$

Cevap

\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}

Adım Adım Çözüm

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.

Anahtar Kavram

Probability of Dependent Events Without Replacement

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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