There are 3 urns. The first contains 3 red and 2 green balls; the second 2 red and 1 green ball; the third 2 red and 4 green balls. A fair die is rolled and the number appearing on top is noted. If the number is odd, the first urn is selected; if the number is 2 or 4 the second urn is selected; if the number is 6 the third urn is selected.
Two balls are drawn at random from the selected urn. If the first ball is green, determine the probability that the second one is red.
I have determined the probability that the first ball is green.
I would just like to know how I work with dependent events like this, it is clearly not Bernoulli trials, but what is the conventional way to do this, or the easiest general way of working with dependent events in this way.
In short notation:
$$P\left(.R\mid G.\right)=\frac{P\left(.R\wedge G.\right)}{P\left(G.\right)}=\frac{P\left(GR\right)}{P\left(G.\right)}$$
Here: $$P\left(G.\right)=P\left(G.\mid\text{odd}\right)P\left(\text{odd}\right)+P\left(G.\mid2\text{ or }4\right)P\left(2\text{ or }4\right)+P\left(G.\mid6\right)P\left(6\right)$$ and:
$$P\left(GR\right)=P\left(GR\mid\text{odd}\right)P\left(\text{odd}\right)+P\left(GR\mid2\text{ or }4\right)P\left(2\text{ or }4\right)+P\left(GR\mid6\right)P\left(6\right)$$