find the probability of probability of randomly selecting three science books and four history books from a box containing five science books and six history books.

Question

zehanz · Answer

This is a counting problem. It has to do with the hypergeometric distribution.
I don't know if you are familaiar with that, so here I go:

You ramdomly pick a number of items (7) out of a total of 11 items. 
There are two kinds of items: s(cience) and h(istory) books.

If X is the number of science books in the 7 books you select, then:

$$P(X=x)=\frac{ \left(\begin{matrix}x \ a\end{matrix}\right)\cdot \left(\begin{matrix}n-x \ b\end{matrix}\right) }{ \left(\begin{matrix}N \ n\end{matrix}\right) }$$

zehanz · Answer

Now this is less scary than you might think!
N = total no of items (11)
n = sample size (7)
a = no of items of the first kind (5 science books)
b = no of items of the second kind (6 history books)
x = no of items of first kind in sample (3 science books)

You want 3 science books and 4 history books, so you can compute the probability as follows:

zehanz · Answer

Sorry, I should have put the a and b on the top in the formula!!

The probability is:$$P(X=3)=\frac{ \left(\begin{matrix}5 \ 3\end{matrix}\right)\cdot \left(\begin{matrix}6 \ 4\end{matrix}\right) }{ \left(\begin{matrix}11 \ 7\end{matrix}\right) }$$

zehanz · Answer

I hope you can follow my explaining!
Also, you still have to calculate the binomial coefficients in the formula.