Conditional probability

Conditional probability

Lessons

\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(AandB)P(A)\frac{P(A\;and \;B)}{P(A)}
  • 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(AandB)P(A)\frac{P(A\;and \;B)}{P(A)}

  • 2.
    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.
    a)
    Find the probability that a red ball is selected.

    b)
    Given that the ball selected is red, find the probability that it came from Bag A.


  • 3.
    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.