The product rule of probability and intuitive explanation I want to make sure I understand the intuition behind the law of product in mathematical probability.
If we have 2 red balls and 5 green balls in a bag and the total number of balls are 20 then the probability of getting a red or a green ball is $\frac{2}{20} + \frac{5}{20} = \frac{7}{20}$ 
This makes sense because the number of red and green balls indicates how likely it is to get one of them.
If we have 3 red balls in a bag with 4 balls total and 2 red balls in a bag with 5 balls total then the probability of getting 2 red balls would be $\frac{3}{4}*\frac{2}{5}=\frac{6}{20}$
Now the reason we do not add the probabilities is because the more red balls we have in the second bag the higher the probability that we get 2 red balls hence we multiply the probability from the first bag to the second bag to show this.  
Is my understanding/explanation correct? How would you describe this intuitively better? 
 A: The key is the same reason America is celebrating tomorrow - independence.
Intuitively, two events are independent if the outcome of one does not affect the other. Since you're drawing from two bags, drawing from one doesn't affect drawing from the other. If you drew from the same bag, you would be drawing without replacement, as the number of objects to select decreases.
Let us consider two cases. First suppose you have one bag with three different coloured marbles, say 3 red, 4 white, 5 blue. What is the probability of selecting a red or blue marble? There are 8 favorable outcomes (the 3 red and the 5 blue) out of 12 total outcomes. The desired probability here is $\frac{8}{12}$. Always associate "or" with addition.
Now what if I'm drawing two marbles? Two examples here.
First, if I just draw a marble and put it back then draw again, I haven't changed the number of outcomes. So here, the probability of drawing two white is $\frac{4}{12}\cdot \frac{4}{12}$.
But if we draw without replacement, we reduce the number of marbles to draw from. The above in this case is $\frac{4}{12}\cdot \frac{3}{11}$.
TL;DR: Associate "and" events with multiplication and "or" events with addition.
A: Intuitively, I find it easier to visualize a tree like, for example, this one :

The number on it are irrelevant but try and imagine the experiment "taking a ball from bag A then one from bag B" as is you were a tiny person walking along the red lines :
First, for bag A, in the picture you have a $3\over7$ chance to go up and pick a red ball (and $1-{3\over7} = {4\over 7}$ chance togo down and pick a Blue one).
Once you done that, you are at another intersection relative to the bag B and you have the same process all over again.
If you have trouble maybe you can also think of obvious counter examples for the addition : Let say you have two bags full of red balls, then $P(\text{ball color = red}) = 1$ for each bag and if you were to add those you would end up with a probability of $2$ which is definitely not correct.
Another trick is to imagine the Bag B (from the picture) is full of red balls (Bag A is still the same as the picture describes it) then it makes sense that not matter what you pick in bag A, you will pick a red ball in bag B with probability $1$ so $P_{\text{Bag A}}(\text{Red}) = P_{\text{Bag A and B}}(\text{Red, Red})$.
In my explanation I'm trying to share some intuitive ways to grasp the concept but Sean answer is more formal and even if my answer helps you might want to look at Sean's and try to understand the key concept of independence/dependence in probability experiments.
Good luck !
A: Well, how about regular multiplication. What are we doing if we say 5x5=25. We are adding 5, 5 times. I.e. 5x5=25=5+5+5+5+5 . Lets think of it in terms of area. We have a 5x5 square, it is made up of 5 1x5 rectangles. Each of those rectangles is made up of 5 1x1 squares. To get the area, we could do a few things. We could simply multiple the lengths of the square (ie 5x5 = 25) or add up the 5 areas of the rectangles (5+5+5+5+5 =25) or add up the areas of all the mini squares (1+1+1+...+1 = 25). See the way the groupings are working, specifically the second one were we add the 5 areas of each rectangle separately to get the area of the large square. 
In your example, there is a 3/4 chance of drawing a red ball from the first bag and 2/5 chance of drawing a red ball from the second bag. 
If the event E is "what is the probability of drawing 2 red balls given the above probabilities" then P(E) is 2/5*3/4 = 3/10.  We are adding 3/4 groups of 2/5 together to get this, much like we were adding 5 groups of 5 together in the area example. 
This isn't intuitive at all, and I don't believe it should be, because we are multiplying by numbers less than 1 and that by itself is just weird to say in English. When I think of multiplication, I think of it as adding M groups of N objects (MxN), this works well when dealing with numbers greater than 1 and less than or equal to zero, but is really weird for numbers in between. I don't believe this is something that is intuitive. However, do think visualizing it with a tree will help. 
