A bag contains 5 red marbles and 5 blue marbles. If 2 marbles are randomly selected from t… — GMAT

A bag contains 5 red marbles and 5 blue marbles. If 2 marbles are randomly selected from the bag one after another without replacement, what is the probability that at least one of the selected marbles is red?

$\frac{7}{9}$ Answer
B
$\frac{2}{9}$
C
$\frac{5}{9}$
D
$\frac{3}{4}$
E
$\frac{1}{4}$

Answer

\frac{7}{9}

To find the probability of drawing at least one red marble, calculate the probability of the complementary event (drawing two blue marbles) and subtract it from 1. The probability of selecting a blue marble on the first draw is

\frac{5}{10}

, and on the second draw without replacement it is

\frac{4}{9}

. The probability of both marbles being blue is

\frac{5}{10} \times \frac{4}{9} = \frac{2}{9}

. Subtracting this from 1 gives

1 - \frac{2}{9} = \frac{7}{9}

Step-by-Step Solution

Identify the complementary outcome

The event 'at least one marble is red' is complementary to the event 'no marbles are red' (i.e., 'both marbles are blue').

Using the relationship

P(\text{at least one red}) = 1 - P(\text{both blue})

is simpler than summing individual favorable cases.

Calculate the probability of drawing two blue marbles without replacement

The probability that the first marble is blue is

\frac{5}{10} = \frac{1}{2}

. The probability that the second marble is blue is

\frac{4}{9}

. Therefore,

P(\text{both blue}) = \frac{5}{10} \times \frac{4}{9} = \frac{20}{90} = \frac{2}{9}

Since draws are without replacement, the sample space and number of blue marbles decrease by 1 after the first draw.

Subtract the complementary probability from 1

P(\text{at least one red}) = 1 - \frac{2}{9} = \frac{7}{9}

The sum of complementary probabilities is always equal to 1.

Key Concept

Complementary Probability:

P(\text{at least one}) = 1 - P(\text{none})

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Key Concept

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