Originally Posted by

**Stormey** Suppose we have a box with 9 balls: 4 blue balls, and 5 red balls.

Two balls are randomly picked from the box.

What would be the probability to draw a blue ball and then a red ball (with NO returning them back)?

So the sample space is $\displaystyle \Omega =\left \{ drawing\hspace{5}any\hspace{5}ball\hspace{5}from \hspace{5} the \hspace{5} box \right \}$, and then $\displaystyle |\Omega |=9$

and in this case, $\displaystyle A=\left \{ drawing\hspace{5}a\hspace{5}blue\hspace{5}ball \right \}$, $\displaystyle B=\left\{ drawing\hspace{5}a\hspace{5}red\hspace{5}ball \right \}$

so if $\displaystyle A$ happened (we drew a blue ball) it'll make the sample space shrink (we're not returning the blue ball back):

$\displaystyle \mathbb{P}(A)=\frac{4}{9}$

and then:

$\displaystyle \mathbb{P}(B)=\frac{5}{8}$

So

$\displaystyle \mathbb{P}(A\cap B)=\frac{4}{9}\cdot\frac{5}{8}$

In the above case, A and B are mutually exclusive (you can't draw blue and red both at the same time, with only one pull), therefore dependent. so far so good.

but, and this is where it gets confusing for me; what happens if there are two independent events?