Multiplication rule for "AND"

Multiplication rule for "AND"

Lessons

\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)
  • 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.

  • 2.
    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?

  • 3.
    Independent Events VS. Dependent Events
    a)
    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

    b)
    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


  • 4.
    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)
    What is the probability of selecting a green ball from Bag A?

    b)
    What is the probability of selecting a green ball?