Hope this isn't a duplicate... have searched but not found my answer, so here goes...
I've been learning about the binomial distribution and was considering a little example when I suddenly realised my intuitive understanding of $P(A \cap B)$ isn't quite right. I'll try and explain below.
I was considering a pretty basic example. I have a bag with 5 balls in it: 3 blue, 2 red. What is the probability of drawing 1 blue ball and 1 red ball if I select using replacement (to make the selections independent).
So... selecting a blue and a red ball. Sounded like $P(blue \cap red)$, which I wrote in shorter hand as $P(B \cap R)$. But, of course, this is not the answer to the question because I can select {RB} or {BR}.
I drew out the sample space to help myself...
There are two clear groups of outcomes that will satisfy the question. I can see that in one group we drew a red ball first and in the other we drew a blue ball first. So, there are $5 \times 5$ total possible events, and of these $6 + 6$ are of interest. Therefore,
$ P(drawing\ a\ blue\ and\ a\ red) = \frac{6}{25} + \frac{6}{25} = 2 \times \frac{6}{25} $
But, this is what confused me... in my intuitive understanding I would have translated $P(drawing\ a\ blue\ and\ a\ red)$ to $P(B \cap R)$, but clearly that would not be correct:
$ P(B \cap R) = P(B) \times P(R) = \frac{3}{5} \times \frac{2}{5} = \frac{6}{25} $
So the two expressions are clearly not the same thing. I'd never really thought of it like this... it is like $P(B \cap R)$ implies an ordering to the selection, which I'd never considered before... blue and red right?! Doesn't seem to suggest I selected a blue then a red in that order: I just selected one of each with no mention of which came first out of the bag...
So, I'm now reasoning it out as follows... is this correct?
If I use conditional probability I can see that $P(A \cap B) = P(A | B)P(B)$ (although, $P(A | B) = P(A)$ because our selections are independent). However, I can see that there appears to be an implied order. But $P(A \cap B) = P(B | A)P(B)$ as well, wich would make sense as we wouldn't expect the order of our selection to alter the overall probability of the outcome. But, clearly, $P(A \cap B)$ implies one particular permutation that leads to the outcome, not every possible permutation that would lead to the outcome.
Is this thinking correct? I never thought of $P(B \cap R)$ as implying a specific order or permutation. Thanks folks :)
This confusion can be resolved by careful attention to definitions and notation.
Where you write $P(B\cap R)$, you call the events $B$ and $R$ "a blue" and "a red", respectively. Implicitly you're referring to two different draws (if you were referring to a single draw, you'd have $P(B\cap R)=0$), but you're not distinguishing the events accordingly, and this leads to confusion.
The events you are interested are $B_a$, a blue ball is drawn on the first draw, $R_a$, a red ball is drawn on the first draw, $B_b$, a blue ball is drawn on the second draw, and $R_b$, a red ball is drawn on the second draw. We have $P(B_a)=P(B_b)=\frac35$ and $P(R_a)=P(R_b)=\frac25$.
You want to know $P((B_a\cap R_b)\cup(B_b\cap R_a))$. Since the events $B_a\cap R_b$ and $B_b\cap R_a$ are mutually exclusive, this is
$$P((B_a\cap R_b)\cup(B_b\cap R_a))=P(B_a\cap R_b)+P(B_b\cap R_a)\;,$$
and since the first and second draws are independent, this is
$$ P(B_a\cap R_b)+P(B_b\cap R_a)=P(B_a)P(R_b)+P(B_b)P(R_a)=2\cdot\frac35\cdot\frac25\;. $$