Q.3.55 - A First Course in Probability

Question

In a class, there are 4 first-year boys, 6 first-year girls, and 6 sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random? 

Step-by-Step Solution

Answer

$9$ sophomore girls must be present, if sex and class are to be independent when a student is selected at random.  

Step 1: Given information

From the information, we observe that a classroom contain 4 first year boys, 6 first year girls and 6 sophomore boys.

We need to find how many sophomore girls must be present if sex and class are to be independent when a student is selected at random.

Step 2: Explanation

Number of boys : $4$

Number of sophomore boys $=6$

Number of sophomore girls $=x$

Then, number of sophomores$=6+x$

Therefore, the total number of students : $4+6+6+x=16+x$

Step 3: Find the probabilities

Let's denote boy as "B" sophomore as "s"

Probability that selected student is a sophomore boy $P\mathit{(}B\mathit{\cap }S\mathit{)}\mathit{=}\frac{\mathit{6}}{\mathit{16}\mathit{+}\mathit{x}}$

Probability that selected student is a sophomore $P\left(S\right)=\frac{6+x}{16+x}$

Probability that selected student is a boy $P\mathit{\left(}B\mathit{\right)}\mathit{=}\frac{\mathit{10}}{\mathit{16}\mathit{+}\mathit{x}}$

For independence, $P(B\cap S)=P\left(B\right) P\left(S\right).$

Step 4: Substitute the probability

Substitute the probability from step 3 in the equation $P(B\cap S)=P\left(B\right) P\left(S\right)$

Then,

$\frac{6}{16+x}=\frac{10}{16+x}\times \frac{6+x}{16+x}$

$\frac{6}{16+x}=\frac{10(6+x)}{{(16+x)}^2}$

$6=\frac{10(6+x)}{16+x}$

Simplify,

$6(16+x)=10(6+x)$

$96+6x=60+10x$
$96-60=10x-6x$

 $36=4x$

Divide

$x=\frac{36}{4}$

$x=9$

Step 5: Final Answer

The number of sophomore girls must be present in the class is 9