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Conditional probability

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Intros
Lessons
  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)}
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Examples
Lessons
  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.
    Topic Notes
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    ⋅\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)}