<s>[INST] A problem Dealing with Sampling with replacement Below is a problem from the Schaum book: "Probability and Statistics". I started
the problem but I am confident that I am on the wrong approach. I am hoping somebody will tell me where I went wrong.
Problem:
An urn holds $60$ red marbles and $40$ white marbles. Two sets of $30$ marbles are drawn with replacement
from the urn, and their color is noted. What is the probability that the two sets differ by $8$ or more red marbles.
Answer:
First we note that both sets should have a mean of $0.4*30 = 7.5$ white marbles. Let $X$ be a random variable
whose value is $1$ if on a single draw the marble is red and $0$ if the marble is white.
\begin{align*}
u_x &= \frac{3}{5} \\
\sigma_x^2 &= E(X^2) - u_x^2 \\
E(X^2) &= \frac{3}{5} \\
\sigma_x^2 &= \frac{3}{5} - \left( \frac{3}{5} \right)^2  = \frac{15}{25} - \frac{9}{25} \\
\sigma_x^2 &= \frac{6}{25} \\
\sigma_s^2 &= \frac{\sigma_x^2}{n} \\
n &= 30 \\
\sigma_s^2 &= \frac{\frac{6}{25}}{30} = \frac{3}{25(15)} \\
\sigma_s^2 &= \frac{1}{125} \\
\sigma_s &= \frac{1}{5\sqrt{5}} \\
\end{align*}
The book's answer to the problem is $0.0482$ and I am confident that my work is wrong. Please tell me where I went wrong.
Thanks,
Bob  
Based upon the response I received, I have been able to continue my work. However, I am stuck. Here is my updated work:
\begin{align*}
u_x &= \frac{3}{5} \\
\sigma_x^2 &= E(X^2) - u_x^2 \\
E(X^2) &= \frac{3}{5} \\
\sigma_x^2 &= \frac{3}{5} - \left( \frac{3}{5} \right)^2  = \frac{15}{25} - \frac{9}{25} \\
\sigma_x^2 &= \frac{6}{25} \\
\sigma_s^2 &= n \sigma_x^2 \\
n &= 30 \\
\sigma_s^2 &= 30 \left( \frac{6}{25} \right) = \frac{6(6)}{5} \\
\sigma_s^2 &= \frac{36}{5} \\
\sigma_s &= \frac{6}{\sqrt{5}} \\
\end{align*}
Let $SD$ be the number of standard deviations $8$ marbles is.
\begin{align*}
SD &= \frac{8}{ \frac{6}{\sqrt{5}}  } = \frac{8\sqrt{5}}{6} \\
SD &= \frac{4 \sqrt{5}}{3} \doteq 2.981424 \\
\end{align*}
At this point, I am not sure what to do.
Bob
 [/INST] The sample variance is $n$ times the variance of one marble.  This makes $\sigma_s^2=\frac {36}{5}$.  The variance of the difference of two samples is twice the variance of one, so $\frac{72}5$.  The square root of this is about $3.79$, so a difference of $8$ is about $2$ standard deviations.
</s>