# 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.
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.