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Multiplication rule for "AND"

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Intros
Lessons
  1. P(A and B) VS. P(A or B)

    P(A and B): probability of event A occurring and then event B occurring in successive trials.
    P(A or B):
    probability of event A occurring or event B occurring during a single trial.
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Examples
Lessons
  1. Multiplication Rule for "AND"
    A coin is tossed, and then a die is rolled.
    What is the probability that the coin shows a head and the die shows a 4?
    1. Independent Events VS. Dependent Events
      1. One card is drawn from a standard deck of 52 cards and is not replaced. A second card is then drawn.
        Consider the following events:
        A = {the 1st1^{st} card is an ace}
        B = {the 2nd2^{nd} card is an ace}
        Determine:
        ⋅\cdot P(A)
        ⋅\cdot P(B)
        ⋅\cdot Are events A, B dependent or independent?
        ⋅\cdot P(A and B), using both the tree diagram and formula
      2. One card is drawn from a standard deck of 52 cards and is replaced. A second card is then drawn.
        Consider the following events:
        A = {the 1st1^{st} card is an ace}
        B = {the 2nd2^{nd} card is an ace}
        Determine:
        ⋅\cdot P(A)
        ⋅\cdot P(B)
        ⋅\cdot Are events A, B dependent or independent?
        ⋅\cdot P(A and B), using both the tree diagram and formula
    2. 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. What is the probability of selecting a green ball from Bag A?
      2. What is the probability of selecting a green ball?
    Topic Notes
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    ⋅\cdot P(A and B): probability of event A occurring and then event B occurring in successive trials.

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

    ⋅\cdot P(A and B) = P(A) ⋅\cdot P(B | A)

    ⋅\cdot Independent Events
    If the events A, B are independent, then the knowledge that event A has occurred has no effect on the probably of the event B occurring, that is P(B | A) = P(B).
    As a result, for independent events: P(A and B) = P(A) ⋅\cdot P(B | A)
    = P(A) ⋅\cdot P(B)