In a class there are 4 freshamn boys, 6 freshman 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?
Hello, r7iris!
Two events, $\displaystyle A$ and $\displaystyle B$, are independent if: .$\displaystyle P(A \cap B) \;=\;P(A)\cdot P(B)$In a class there are 4 freshman boys, 6 freshman 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?
Let $\displaystyle x$ = number of sophomore girls.
Tabulate the information:
$\displaystyle \begin{array}{cccccccc} & | & \text{Frosh} & | & \text{Soph} & | & \text{Total} & | \\ \hline
\text{Boys} &|& 4 &|& 6 &|& 10 & | \\ \hline
\text{Girls} &|& 6 &|& x &|& x+6 & | \\ \hline
\text{Total} &|& 10 &|& x+6 &|& x+16 & |
\end{array}$
We have: . $\displaystyle P(\text{Boy}\cap\text{Frosh}) \:=\:\frac{4}{x+16}\qquad P(\text{Boy}) \:=\:\frac{10}{x+16} \qquad P(\text{Frosh}) \:=\:\frac{10}{x+16}$
If $\displaystyle P(\text{Boy})$ and $\displaystyle P(\text{Frosh})$ are independent,
. . then: .$\displaystyle P(\text{Boy}\cap\text{Frosh}) \;=\;P(\text{Boy})\cdot P(\text{Frosh})$
So we have: .$\displaystyle \frac{4}{x+16} \;=\;\frac{10}{x+16}\cdot\frac{10}{x+16}$
Multiply by $\displaystyle (x+16)^2\!:\;\;4(x+16) \:=\:100\quad\Rightarrow\quad x \,=\,9$
Therefore, there must be 9 sophomore girls.