Question

From a group of $3$first-year students,$4$sophomores, $4$juniors, and$3$seniors, a committee of size $4$is randomly selected. Find the probability that the committee will consist of

$\left(a\right)1$from each class;

$\left(b\right)2$sophomores and $2$juniors;

$\left(c\right)$only sophomores or juniors.

Short answer

$a\left)P\right(1$from each class$)=\frac{144}{\left(\begin{array}{c}14\\ 4\end{array}\right)}\approx 14.4\%$

$b)P$($2$sophomores, $2$juniors)$\cong 3.6\%$

$c)P($only sophomores and juniors$)=\frac{\left(\begin{array}{l}8\\ 4\end{array}\right)}{\left(\begin{array}{c}14\\ 4\end{array}\right)}\approx 7.0\%$

Step-by-step solution

Given Information.

A group of $3$first-year students, $4$sophomores, $4$juniors, and$3$seniors, a committee of size $4$is randomly selected.

Part (a) Explanation.

$P$(the chosen committee has one student from each class)

Count the number of elements in this event (which are members of$S$).

There are$3$choices for a freshman,$4$choices for a sophomore, $4$choices for a junior, and$3$for a senior.

There are$3·4·4·3$different choices for a committee with$1$students from each class

$P\left(1\text{from each class}\right)=\frac{144}{\left(\begin{array}{c}14\\ 4\end{array}\right)}\approx 0.144$

There is approximately a $14.4\%$chance that the committee has a student from each class.

Part (b) Explanation.

$P$(committee consists of $2$sophomores and $2$juniors)

Count the number of elements $S$that are in this event.

First, choose $2$sophomores out of $4$.$\left(\begin{array}{l}4\\ 2\end{array}\right)$ways.

Then, choose $2$juniors out of$4.\left(\begin{array}{l}4\\ 2\end{array}\right)$ways.

These $\left(\begin{array}{l}4\\ 2\end{array}\right)·\left(\begin{array}{l}4\\ 2\end{array}\right)$choices define every event from the outcome space that includes$2$sophomores and$2$juniors.

$P\left(2\text{sophomors, 2juniors}\right)=\frac{\left(\begin{array}{l}4\\ 2\end{array}\right)·\left(\begin{array}{l}4\\ 2\end{array}\right)}{\left(\begin{array}{c}14\\ 4\end{array}\right)}\approx 0.036$

There is approximately a $3.6\%$chance that the chosen committee consists of $2$sophomores and $2$juniors.

Part (c) Explanation.

$P$(the committee includes only sophomores and juniors)

Count the number of elements from $S$which only sophomores and juniors are chosen.

there are $8$sophomores and juniors together, and there are$\left(\begin{array}{l}8\\ 4\end{array}\right)$committees $4$chosen among those$7$people.

$P\left(\text{only sophomores and juniors}\right)=\frac{\left(\begin{array}{l}8\\ 4\end{array}\right)}{\left(\begin{array}{c}14\\ 4\end{array}\right)}\approx 0.070$

There is approximately a $7.0\%$chance that the committee consists of only sophomores and juniors.