Q. 2.8 - A First Course in Probability

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.

Step-by-Step Solution

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 1 Given Information.

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

Step 2 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.

Step 3 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.

Step 4 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.