A committee of 10 members consists of 5 senior directors and 5 junior associates. If 3 com… — GMAT

A committee of $10$ members consists of $5$ senior directors and $5$ junior associates. If $3$ committee members are selected at random one after another without replacement, what is the probability that exactly $2$ of the selected members are senior directors?

A
$\frac{5}{36}$
B
$\frac{3}{8}$
$\frac{5}{12}$ Answer
D
$\frac{25}{144}$
E
$\frac{1}{2}$

Answer

The probability that exactly

2

of the selected members are senior directors is

\frac{5}{12}

The correct answer is found by taking into account that the selections are dependent events without replacement. The probability of selecting senior directors on the first two draws and a junior associate on the third is

\frac{5}{10} \times \frac{4}{9} \times \frac{5}{8} = \frac{5}{36}

. Because there are

3

possible mutually exclusive orders to select

2

senior directors and

1

junior associate, multiplying

\frac{5}{36}

3

gives the final probability of

\frac{5}{12}

Step-by-Step Solution

Calculate the probability of drawing Senior, Senior, Junior in that specific order without replacement.

P(\text{Senior}_1) \times P(\text{Senior}_2 \mid \text{Senior}_1) \times P(\text{Junior}_3 \mid \text{Senior}_1 \cap \text{Senior}_2) = \frac{5}{10} \times \frac{4}{9} \times \frac{5}{8} = \frac{100}{720} = \frac{5}{36}

Because draws are made without replacement, the sample space and favorable outcomes decrease with each draw.

Determine the number of distinct orderings of

2

Senior directors and

1

Junior associate.

\binom{3}{2} = 3

orderings: (Senior, Senior, Junior), (Senior, Junior, Senior), and (Junior, Senior, Senior).

The question asks for exactly

2

senior directors regardless of the order in which they are selected.

Multiply the single-sequence probability by the total number of valid orderings.

3 \times \frac{5}{36} = \frac{15}{36} = \frac{5}{12}

Since each distinct ordering is mutually exclusive and has the same probability, the total probability is the sum across all

3

orderings.

Key Concept

Probability of Dependent Events (Sampling Without Replacement)

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