A high school student council must select a 3-member executive committee consisting of a P… — ACT

A high school student council must select a 3-member executive committee consisting of a President, a Vice President, and a Treasurer from a group of 6 juniors and 4 seniors. If the positions are filled sequentially at random and no student can hold more than one position, what is the probability that the President is a senior, the Vice President is a junior, and the Treasurer is a senior?

$\frac{1}{10}$ Answer
B
$\frac{3}{10}$
C
$\frac{2}{15}$
D
$\frac{12}{125}$
E
$\frac{17}{120}$

Answer

The probability that the President is a senior, the Vice President is a junior, and the Treasurer is a senior is

\frac{1}{10}

To find the probability of dependent sequential events, multiply the probability of each event given that the preceding events have occurred. First, selecting a senior President has a probability of

\frac{4}{10}

. Next, selecting a junior Vice President from the remaining 9 students has a probability of

\frac{6}{9}

. Finally, selecting another senior as Treasurer from the remaining 8 students (which now contains 3 seniors) has a probability of

\frac{3}{8}

. Multiplying these probabilities yields

\frac{4}{10} \times \frac{6}{9} \times \frac{3}{8} = \frac{72}{720} = \frac{1}{10}

Step-by-Step Solution

Determine the probability of choosing a senior as President.

There are 4 seniors out of 10 total students, so

P(\text{President is Senior}) = \frac{4}{10}

The first selection is made from the full group of 10 students.

Determine the probability of choosing a junior as Vice President given the first event.

There are 6 juniors remaining out of 9 total remaining students, so

P(\text{VP is Junior} \mid \text{President is Senior}) = \frac{6}{9}

One student (a senior) has already been selected, reducing the total count to 9.

Determine the probability of choosing a senior as Treasurer given the first two events.

There are 3 seniors remaining out of 8 total remaining students, so

P(\text{Treasurer is Senior} \mid \text{first two selections}) = \frac{3}{8}

Two students have now been selected, leaving 8 total students and 3 remaining seniors.

Multiply the sequential conditional probabilities together.

\frac{4}{10} \times \frac{6}{9} \times \frac{3}{8} = \frac{72}{720} = \frac{1}{10}

According to the Multiplication Rule for Dependent Events, the joint probability is the product of the sequential probabilities.

Key Concept

Probability of Dependent Compound Events

Estimated Time:1m 30s

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