A box contains 4 red balls and 6 blue balls. If two balls are randomly drawn from the box … — GMAT

A box contains $4$ red balls and $6$ blue balls. If two balls are randomly drawn from the box one after another without replacement, what is the probability that both balls drawn are red?

A
$\frac{4}{25}$
B
$\frac{3}{25}$
$\frac{2}{15}$ Cevap
D
$\frac{1}{3}$
E
$\frac{11}{15}$

Cevap

The probability that both balls drawn are red is

\frac{2}{15}

The probability of selecting a red ball on the first draw is

\frac{4}{10}

. Since the ball is not replaced,

9

total balls remain in the box, of which

3

are red. The probability of selecting a red ball on the second draw given the first was red is

\frac{3}{9}

. By the multiplication rule for dependent events, the probability that both balls drawn are red is

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

Adım Adım Çözüm

Find the probability of drawing a red ball on the first pick.

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

There are

4

red balls out of a total of

10

balls (

4 + 6 = 10

Find the conditional probability of drawing a red ball on the second pick after one red ball has been removed.

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

Because sampling is done without replacement,

1

red ball and

1

total ball are removed, leaving

3

red balls out of

9

remaining balls.

Multiply the probabilities of the dependent events to find the joint probability.

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

For dependent events,

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

Anahtar Kavram

Probability of Dependent Events (Sampling Without Replacement)

Soru

Cevap

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Anahtar Kavram

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