Conditional probability

  1. 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) = P(A  and  B)P(A)\frac{P(A\;and \;B)}{P(A)}
  1. 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.
  2. 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.