A bakery display case contains 8 fruit tarts and 12 chocolate croissants. If a customer ra… — GMAT

A bakery display case contains $8$ fruit tarts and $12$ chocolate croissants. If a customer randomly selects $2$ pastry items one after another without replacement, what is the probability that the customer chooses exactly $1$ fruit tart and $1$ chocolate croissant?

A
$\frac{24}{95}$
B
$\frac{12}{25}$
$\frac{48}{95}$ Answer
D
$\frac{47}{95}$
E
$\frac{6}{25}$

Answer

The probability that the customer chooses exactly

1

fruit tart and

1

chocolate croissant is

\frac{48}{95}

Selecting one fruit tart and one chocolate croissant can occur in two mutually exclusive orderings: (Fruit Tart, Chocolate Croissant) or (Chocolate Croissant, Fruit Tart). Since the selection is without replacement, the total number of remaining items drops to

19

for the second draw. The probability of the first order is

\frac{8}{20} \times \frac{12}{19} = \frac{96}{380}

, and the probability of the second order is

\frac{12}{20} \times \frac{8}{19} = \frac{96}{380}

. Summing these gives

\frac{192}{380}

, which simplifies to

\frac{48}{95}

Step-by-Step Solution

Determine the total number of items and individual counts.

Total items =

8 + 12 = 20

. Fruit tarts =

8

, Chocolate croissants =

12

Establishing the initial sample space is necessary for calculating single-event probabilities.

Calculate the probability for each distinct order of selection without replacement.

Probability of (Fruit Tart then Chocolate Croissant) =

\frac{8}{20} \times \frac{12}{19} = \frac{96}{380}

. Probability of (Chocolate Croissant then Fruit Tart) =

\frac{12}{20} \times \frac{8}{19} = \frac{96}{380}

Because sampling is done without replacement, the total pool decreases from

20

19

after the first selection.

Sum the probabilities of the mutually exclusive outcomes and simplify.

\frac{96}{380} + \frac{96}{380} = \frac{192}{380} = \frac{48}{95}

The desired event occurs if either sequence happens, so the probabilities of the two sequences are added together.

Key Concept

Probability of Dependent Events (Sampling Without Replacement)

Estimated Time:1m 30s

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