Statistics Help

How do you figure out what the probability is that two things will happen at the same time? Consider the following:

You've got a bag full of 100 marbles. In this bag are 60 red marbles (that's 3/5 of 100), and 40 green marbles (the final 2/5).

In addition, some of the marbles are "glassies" (solid glass color) and some are "catseyes" (with a flash of white in each marble). 1/5 are catseyes; the rest are glassies. And the proportion of glassies to catseyes is independent of the proportion of red to green marbles; in other words, there's just as much chance that a red marble is a catseye as there is that a green marble will be a catseye.

So if you reach into the bag without looking and pull out one marble, the chance that it will be red is 3/5. The chance that it will be green is 2/5.

And if you reach into the bag and randomly pull out a marble, there's a 1/5 chance that it will be a catseye, and a 4/5 chance that it will be a glassy.

OK, this is pretty simple. But here's a harder problem. If you reach into the bag and pull out a single marble at random, what is the chance that it will be both green and a catseye?

Well, we've already said that these two marble qualities are independent of each other in our bag, so what that means is that, of all the green marbles, 1/5 of them are catseyes. But only 2/5 of the total marbles are green. So that means that the green catseyes are 1/5 of 2/5 of the total marbles. You calculate this like so:

2/5 (green) x 1/5 (catseye) = 2/25 (green and catseye)

You can figure out the other combinations, too.

For red catseyes: 3/5 (red) x 1/5 (catseye) = 3/25 (red and catseye)

For green glassies: 2/5 (green) x 4/5 (glassy) = 8/25 (green and glassy)

And finally, for red glassies: 3/5 (red) x 4/5 (glassy) = 12/25 (red and glassy)

Note that 2/25 + 3/25 + 8/25 + 12/25 = all 25 25ths. They have to add up to all of our marbles, right? It would be too bad to lose some LOL!

Oh, and also note that this only works if the two characteristics are truly independent of each other. If there's a greater chance for a red marble to be a catseye than there is for a green marble, this messes up our neat statistics.