Conditional probability

Definition of Conditional Probability
$\cdot$ P(B | A): probability of event B occurring, given that event A has already occurred.

$\cdot$ recall: P(A and B) = P(A) $\cdot$ P(B | A)

then: P(B | A) = $\frac{P(A\;and \;B)}{P(A)}$

Probability Tree Diagram
Bag A contains 2 red balls and 3 green balls. Bag B contains 1 red ball and 4 green balls.
A fair die is rolled: if a 1 or 2 comes up, a ball is randomly selected from Bag A;

if a 3, 4, 5, or 6 comes up, a ball is randomly selected from Bag B.
1. Find the probability that a red ball is selected.
2. Given that the ball selected is red, find the probability that it came from Bag A.
It is known that 60% of graduating students are girls. Two grads are chosen at random. Given that at least one of the two grads are girls, determine the probability that both grads are girls.